Download Chapter 2. Algebra

Chapter 2. Algebra
Chapter 2: Algebra
Unit 1: EQUATIONS AND INEQUALITIES
____________________________________________
1. VOCABULARY AND GRAMMAR REVIEW:
1.1 Vocabulary:
- Abstract algebra (modern algebra): (n) đại số hiện đại
- Acceleration (n): gia tốc
- Algebra (n): đại số
- Algebraic (a) thuộc về đại số
- Algebraic combinatorics (n): đại số tổ hợp
- Algebraic expression (n): biểu thức đại số
- Algebraic geometry (n): Hình học đại số
- Algebraic structure (n): cấu trúc đại số
- Analysis (n): giải tích
- Binomial (n): nhị thức
- Coefficient (n): hệ số
- Conditional inequality (n): bất phương trình có điều kiện
- Constant (n): hằng số
- Coordinates (n): Tọa độ
- Cubic equation (n): phương trình bậc 3
- Discriminant (n): biểu thức delta của phương trình bậc 2
- Elementary algebra (n): đại số sơ cấp
- Fields (n): trường
- Geometry (n): hình học
- Gravity (n): trọng lực
- Group (n): nhóm
- Half-plane (n): nửa mặt phẳng
- Inequality (n): bất phương trình
- Intercept (n): phần mặt phẳng hoặc đường thẳng bị chắn.
- Linear algebra (n): đại số tuyến tính
- Manipulate (v): vận dụng
- Mass (n) khối, khối lượng
- Mathematical sentence (n): Mệnh đề toán học
- Monomial (n): nhị thức
- Number theory (n): lý thuyết số
- Oscillation (n): sự dao động
- Pendulum (n) con lắc
- Perpendicular (a): vuông góc
- Point of intersection (n): giao điểm
- Polynomial (n): đa thức
- Pure mathematics (n): Toán học thuần túy
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Chapter 2. Algebra
- Quadratic equation (n): phương trình bậc hai
- Reciprocal (n): phần tử nghịch đảo
- Reflexive (a): phản xạ
- Ring (n): vành
- Slope (n): hệ số góc (độ dốc của đường thẳng)
- Statement (n): mệnh đề
- Symbol (n): ký hiệu
- Symmetric(a): đối xứng
- Term (n): số hạng
- Topology (n): tôpô
- Transitive (a): bắt cầu
- Unconditional inequality (n): bất đẳng thức luôn đúng
- Universal (a): phổ biến, phổ quát
- Unknown (n): biến, đại lượng chưa biết.
- Variable (n): biến
1.2 Grammar review:
- Tenses
- Comparison
- If clauses
- V-ing and to infinitive.
- Passive – Active voices.
1.3 Exercises:
1.3.1 Choose the correct word to fill in the blanks:
together collection expression coefficients constant
terms
symbols
An algebraic expression is a ____(1)___ of variables, constants and numbers connected
____(2)___ by the arithmetic operations. For example, 3  a( x  y )2 is an algebraic ___(3)___
involving the variables x and y, the ____(4)___ a and the number 3, 2. It consists of terms and
____(5)____. The terms are symbols or group of ___(6)____ and the coefficients are the
numbers that accompany the ___(7)___. For example, with the above algebraic expression a is
2
the coefficient of the  x  y  term.
1.3.2 Choose the suitable word to fill in the blank:
Monomial
Variable
Inequality
Polynomial Equation Binomial
Quadratic equation
Coefficient
a) ________: is a letter used to represent an unknown quantity.
b) ________: is the numerical part of an algebraic term.
c) ________: is an algebraic expression made up only one term, for example: 3ab
d) ________: is an algebraic expression made up of two terms separated by a plus or minus
sign, for example: x  5; y 2  y .
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Chapter 2. Algebra
e) ________: is an algebraic expression made up of two or more terms separated by plus or
minus signs, for example: x3  4;4n4  3n3  n .
f) ________: is a statement of equality between two algebraic expressions, such as:
5x  20  x 2 ; a3  a  3
g) ________ involves variables that are squared but none that are raised to higher power.
h) ________ is a statement that two algebraic expressions are not equal or that one is greater
than or equal to or less than or equal to another.
1.3.3 Fill in the blanks with the suitable form of the words in the brackets
The equation of a straight line y = ax + b contains four quantities. These are two
___(1)___(vary) x and y, whose values are the coordinates of a point lying on the line, and two
constants. The constants are the gradient (or slope) a and the vertical intercept b.
_____(2)___(find) the value of any one of these quantities we require the values of the other
three. The value of the fourth is then obtained by balancing the equation. A line with a positive
slope (a > 0) ____(3)____(rise) from left to right, while one with negative slope (a < 0) falls from
left to right. A line with a slope of zero (a = 0) is ____(4)___(horizontally) and with an undefined
slope (a is undefined) is vertical.
An alternative form of the straight line is px + qy = r, where p, q and r are constants. In this
form the gradient ____(5)___(give) by –p/q and the intercept by r/q. For example, the equation:
3x + 4y = 5 represents the straight line, with the gradient is (-3)/4 and the intercept is 5/4. The
two lines ___(6)___(be) parallel if they have the same gradient but different vertical intercept. If
two lines intersect to form a right triangle, they ____(7)___(call) perpendicular lines. Therefore,
the lines are perpendicular if and only if their slopes are negative reciprocals of each other.
1.3.4 Fill in the blanks with the correct form of the verb in the bracket
If an inequality is _____(1)___(to write) in the form f (x) > 0, then by constructing the graph
of the function y = f(x) one can ___(2)____(to see) at a glance values of which the inequality is
satisfied (the graph lies above the x – axis). The solution is exact or approximate depending on
whether the points where the graph ____(3)___(to pass) from the lower half-plane y < 0 into the
upper half-plane y > 0 ____(4)___(to be) exact or approximate.
If the inequality is ____(5)___(to give) in the form f1 ( x)  f 2 ( x) , it is possible to construct the
graphs of two functions y  f1 ( x) and y  f 2 ( x) , then to determine from the drawing those values
of x for which the first graph lies above the second one. The set of such values of x will then
yield the solution set of the inequality.
The basic valuable feature of the graphical approach to solving inequalities is that even a
rough graph of the function very often shows that the inequality ____(6)___(to hold) true in the
intervals bounded by such characteristic points as the points of intersection of the graph
y  f1 ( x) and y  f 2 ( x) (or the point of intersection of the graph y = f (x) with the x-axis). Finding
these points ____(7)___(to be) a somewhat simpler job because it ___(8)___(to reduce) to
solving equations and not inequalities.
1.3.5 Put the words in the following sentences in the correct order:
a) Variable/ unknown / is / a / letter / an /used / to / represent / quantity.
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Chapter 2. Algebra
b) Two / straight / lines / are /either / parallel / or / they / intersect / at / a / point.
c) An / equal / is / equation / a / that / says statement / / two / algebraic / expressions / are.
d) An / and / coefficients algebraic / expression / consists / of / terms /.
e) The / symbols / terms / symbols / are / or / group / of.
f) The / that / accompany / are / the / coefficients / numbers / the / terms.
g) An / equation / two / equations / is / a / statement / of / the / equality / of.
h) A / involving / sums / of / polynomial / is / an / expression / powers / of / a / variable.
1.3.6 Find and the correct the mistake in the following sentences:
a) An algebraic expression consist of terms and coefficients.
A
B
C
D
b) Expression can been simplified by combining like terms and by factorizing similar terms.
A
B
C
D
E
c) A equation is a statement of the equality of two expressions.
A
B
C
D
d) A polynomial is a expression involving sums of powers of a variable.
A
B
C
D
e) The coefficients are the numbers that accompanies the terms.
A
B
C
D
2. READING
2.1 ALGEBRA
Algebra is the branch of mathematics concerning the study of the rules of operations and
relations, and the constructions and concepts arising from them, including terms, polynomials,
equations and algebraic structures. Together with geometry, analysis, topology, combinatorics,
and number theory, algebra is one of the main branches of pure mathematics.
Elementary algebra is often part of the curriculum in secondary education and introduces the
concept of variables representing numbers. Statements based on these variables are manipulated
using the rules of operations that apply to numbers, such as addition. This can be done for a
variety of reasons, including equation solving. Algebra is much broader than elementary algebra
and studies what happens when different rules of operations are used and when operations are
devised for things other than numbers. Addition and multiplication can be generalized and their
precise definitions lead to structures such as groups, rings and fields, studied in the area of
mathematics called abstract algebra. These are some parts of algebra such as:
 Abstract algebra, sometimes also called modern algebra, in which algebraic structures
such as groups, rings and fields are axiomatically defined and investigated.
 Linear algebra, in which the specific properties of vector spaces are studied (including
matrices);
 Universal algebra, in which properties common to all algebraic structures are studied.
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Chapter 2. Algebra

Algebraic number theory, in which the properties of numbers are studied through
algebraic systems. Number theory inspired much of the original abstraction in algebra.
 Algebraic geometry applies abstract algebra to the problems of geometry.
 Algebraic combinatorics, in which abstract algebraic methods are used to study
combinatorial questions.
In some directions of advanced study, axiomatic algebraic systems such as groups, rings,
fields, and algebras over a field are investigated in the presence of a geometric structure (a metric
or a topology) which is compatible with the algebraic structure.
In fact, Elementary algebra is the most basic form of algebra. It is taught to students who are
presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. In
arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur. In algebra,
numbers are often denoted by symbols (such as a, x, or y). This is useful because:
 It allows the general formulation of arithmetical laws (such as a + b = b + a for all a and
b), and thus is the first step to a systematic exploration of the properties of the real number
system.
 It allows the reference to "unknown" numbers, the formulation of equations and the study
of how to solve these (for instance, "Find a number x such that 3x + 1 = 10" or going a bit further
"Find a number x such that ax+b=c". Step which lets to the conclusion that is not the nature of the
specific numbers the one that allows us to solve it but that of the operations involved).
 It allows the formulation of functional relationships (such as "If you sell x tickets, then
your profit will be 3x − 10 dollars, or f(x) = 3x − 10, where f is the function, and x is the number
to which the function is applied.").
In this chapter, we focus in some subjects of elementary algebra such as: function, equation,
or inequality and some articles about linear algebra and abstract algebra.
Comprehension check:
A) Answer the following question:
1) According to the text, what is algebra?
2) What are the branches of mathematics?
3) What are the parts of elementary algebra?
4) What is modern algebra about?
5) What does algebraic geometry do?
6) Why are numbers usually denoted by symbol in algebra?
7) What are the parts of algebra that you have studied? Which is the most difficult? Which is
the part that you interest most?
2.2 EQUATION:
An equation is a statement that says two algebraic expressions are equal. If we assign a
specific number to each variable or unknown in an algebraic expression, then the algebraic
expression will be equal to a number. This is called evaluating the expression.
If we evaluate each side of an equation and the number obtained is the same for each side of
the equation, then the specific values assigned to the unknowns are called a solution of the
equation. However, some equations do not have any solutions that are real numbers.
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Chapter 2. Algebra
One equation is equivalent to another equation if the two have exactly the same solutions. The
basic idea in solving equations is to transform a given equation into an equivalent equation
whose solutions are obvious. The two main tools for solving equations are:
1) If you add or subtract the same algebraic expression to or from each side of an equation,
the resulting equation is equivalent to the original equation.
2) If you multiply or divide both sides of an equation by the same nonzero algebraic
expression, the resulting equation is equivalent to the original equation.
If the equation involves square roots, then you can solve it by squaring each side of the
equation. However, you should remember to check your answer since squaring each side does
not always give an equivalent equation.
Comprehension check:
Answer the following question:
1) What is an equation? Give an example of equation?
2) What do we do to evaluate an algebraic expression?
3) What is the solution of an equation? Are there any types of equations that don’t have
solutions? Why?
4) What are equivalent equations?
5) What shall we do to solve an equation?
6) What shall we do to solve the equation involves square roots?
2.3 POLYNOMIAL EQUATIONS
A polynomial expression equated to zero is called a polynomial equation. The general form
of a polynomial expression is an x n  an 1 x n 1  ....  a1 x  a0  0 , where an  0 . There are a number
of specific polynomial equations of interest.
A linear equation is any equation of the form ax + b = 0. The graph associated with a linear
expression is a straight line. The point on the line that corresponds to the value of the expression
being zero is where the line crosses the x – axis. The value of x at the point where the line crosses
the x – axis is called the solution of the equation.
A quadratic equation is any equation of the form ax 2  bx  c  0 . A quadratic expression has
an associated graph in the form of a parabola. Just as with the linear expression, the quadratic
expression has the value 0 when the graph – the parabola crosses the x – axis. This means that the
solution to the quadratic equation can consist of 2, 1 or no real number values of x.
Cubic equation is any equation of the form ax3  bx 2  cx  d  0 , where a  0 . The cubic
equation has a solution that may consist of 1, 2, or 3 x – values. Note that the cubic always has at
least one real value of x as a solution because the cubic expression has values ranging from
negative to positive or from positive to negative. Consequently, there must be at least one value
of x where the curve crosses the x – axis.
A) Choose True (T) or False (F):
a) ________ A linear equation always has a real solution.
b) _________ A quadratic equation always has at least one real solution.
c) _________Cubic equation may have more than 3 real solutions.
d) _________ The graph of cubic expression always crosses the x-axis.
e) _________ The graph associated with a linear expression is a parabola.
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Chapter 2. Algebra
f) ________ The cubic equation always has at least one real solution.
g) ________ The graph associated with a cubic equation is always a parabola.
2.4 SOMETHING ABOUT MATHEMATICIAL SENTENCES
A mathematical sentence containing an equal sign is an equation. The two parts of an
equation are called its members. A mathematical sentence that is either true or false but not both
is called a closed sentence. To decide whether a closed sentence containing an equal sign is true
or false, we check to see that both elements, or members of the sentence name the same number.
To decide whether a closed sentence containing an  sign is true or false, we check to see that
both elements do not name the same number.
The relation of equality between two numbers satisfies the following basic axioms for the
number a, b and c.
Reflexive: a = a.
Symmetric: If a = b then b = a
Transitive: If a = b and b = c then a = c.
While the symbol = in an arithmetic sentence means is equal to, another symbol  , mean is
not equal to. When an = sign is replaced by  sign, the opposite meaning is implied. (Thus 8 =
11 – 3 is read eight is equal to eleven minus three while 9  6  13 is read nine plus six is not equal
to thirteen.)
The important feature about a sentence involving numerals is that it is either true or false, but
not both. There is nothing incorrect about writing a false sentence, in fact in some mathematical
proofs it is essential that you write a false sentence.
We already know that if we draw one short line across the symbol = we change it to  . The
symbol  implies either of two things – is greater than or is less than. In other words the sign 
in 3  4  6 tells us only that numerals 3 +4 and 6 name different numbers, but does not tell us
which numeral names the greater or the lesser of the two numbers.
To know which of the two numbers is greater let us use the conventional symbol < and >, <
means is less than, while > means is greater than. These are inequality symbols because they
indicate order of numbers. (6 < 7 is read six is less than seven, 29 > 3 is read twenty nine is
greater than three). The signs which express equality or inequality (, , , ) are called relation
symbols because they indicate how two expressions are related. Moreover, we sometimes use the
symbols  (  ) means is greater than or equal (is smaller than or equal).
Comprehension check:
A) Answer the following questions:
1) How can we create an equation from a mathematical sentence?
2) What are the basic axioms for the number a, b and c?
3) What should we do to check whether a closed sentence containing an equal sign is true or
false?
4) What is the important feature about a sentence involving numerals?
5) Which is the symbol  implies?
B) Write down these expressions in formula:
1) T is greater than or equal two times Z plus 4.
2) Square x is equal to 5 times x minus 6.
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Chapter 2. Algebra
3) y is equal square roof of x plus 6.
4) Two times y is less than three times cubed x.
C) Express the symbol (, , , ) in arithmetic sentences:
Example: a > b : a is greater than b
a) a  b 2
b)   
c) a 2  2ab  b  x 2  y 2
d) x 3  x  1
e) x  y  2
2.5 INEQUALITIES
An inequality is simple a statement that one expression is greater than or less than another.
We have seen the symbol a > b, which reads “a is greater than b” and a < b, which reads “a is less
than b”. There are many ways in which to make these statements. For example, there are three
ways of expressing the statement “a is greater than b”.
 a < b or b < a
 a – b > 0; a – b is a positive number.
 a – b = n, n is a positive number.
If an expression is either greater than or equal to, we use the symbol  , and similarly, 
states is less than or equal to. Two inequalities are alike in sense or of the same sense, if their
symbols for inequality point in the same direction. Similarly, they are unalike, or opposite in
sense, if the symbols point in opposite directions.
In discussing inequalities of algebraic expressions we see that we can have two classes of
them:
1. If the sense of inequality is the same for all values of the symbols for which its member
are defined, the inequality is called an absolute or unconditional inequality
Illustrations: x 2  y 2  0, x  0 , and y  0
 4
2. If the sense of inequality holds only for certain values of the symbols involved, the
inequality is called a conditional inequality
Illustrations: x+ 3 < 7, true only for values of x less than four;
x 2  6  5 x , true only for x between 2 and 3.
The inequality symbols are frequently used to denote the values of a variable between given
limits. Thus, 1  x  4 , states “values of x from 1, including 1, to 4 but not including 4”, i.e., x
may assume the value 1 and from 1 to 4 but no others. This is also called “defining the range of
values”.
x 2  6  5 x for 2  x  3 .
Properties:
a. The sense of an inequality is not changed if both members are increased or decreased by
the same number.
If a > b then a + x > b + x and a – x > b – x.
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Chapter 2. Algebra
a b

x x
a b
c. If a > b and x < 0 then ax < bx and 
x x
b. If a > b and x > 0 then: ax > bx and
d. If a, b and n are positive numbers and a > b, then: a n  b n and n a  n b
e. If x > 0, a > b and a, b have like signs then
x x

a b
We can illustrate these properties by using numbers.
Illustrations:
(1) Since 4 > 3, we have 4 +2 > 3 + 2 as 6 > 5
(2) Since 4 > 3, we have 4(2) >3(2) as 8 > 6
(3) Since 4 > 3, we have 4(-2)<3(-2) as -8 < -6
(4) Since 16>9, we have 16  9 as 4 > 3
(5) Since 4 > 3, we have
2 2
1 2
 as 
4 3
2 3
The solutions of inequalities are obtained in a manner very similar to that of obtaining
solutions to equations. The main difference is that we are now finding a range of values of the
unknown such that the inequality is satisfied. Furthermore, we must pay strict attention to the
properties so that in performing operations we do not change the sense of inequality without
knowing it.
Comprehension check
Answer the following questions
a. What is an inequality in maths?
b. What does the following mean a > b?
c. According to the text, how many classes of inequality? What are they?
d. Which symbol do we use to signify an expression “is either greater than or equal to”?
e. When are two inequalities like or unlike in sense?
f. Which symbol do we use to signify an expression “is either less than or equal to”?
g. What are the properties of inequality?
Do you know any kinds of inequality?
3. SPEAKING – WRITING – LISTENING - DISCUSSION
3.1 Discussion:
a) In your opinion, what should we do to be good at algebra?
b) List some types of equations you have ever learned? Give examples for each type of
equations.
c) How to solve a quadratic equation and cubic equation; quintic equation? Give examples.
d) List some types of inequalities you have ever learned, give examples.
3.2 Writing:
Put these sentences in the correct order
____(1)___In order to communicate effectively, we must agree on the precise meaning of the
terms which we use.
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Chapter 2. Algebra
_______It’s necessary to define all terms to be used.
_______However, it is impossible to do this since to define a word we must use others words
and thus circularity cannot be avoided.
________In mathematics, we choose certain terms as undefined and define the others by
using these terms.
________Similarly, as we are unable to define all terms, we cannot prove the truth of all
statements.
________ Such statements which are assumed to be true without proof are called axioms.
________ Thus we must begin by assuming the truth of some statements without proof.
________ The work of a mathematician consists of proving that certain sentences are terms,
theorem already proved, and some laws of logic which have been carefully laid down….
________ Sentences which are proved to be laws are called theorems
3.3 Listening:
Listen to the tape and fill in the blanks:
An____(1)____expression is a____(2)____ of variables, constants and____(3)___ connected
together by the arithmetic operations. It consists of____(4)___ and coefficients. The terms
are____(5)____or group of symbols and the coefficients are the numbers that____(6)___ the
terms. Algebraic expressions can be____(7)_____by combining like terms and
by____(8)____similar terms. Similar terms are____(9)____ with a symbol in common and
common symbols can be factorized. This____(10)___ is called factorizing the expression.
Sometimes it may be____(11)___ to reverse the process of factorizing an expression
by___(12)___the brackets. This process is known as expanding the brackets. Moreover,
expressions can be evaluated by____(13)____ numbers for symbols or even by substituting
alternative symbols.
4. TRANSLATION:
4.1 Translate into English:
Translate the following sentences into English:
Nếu đa thức f(x) chia cho x – k, thì số dư là r = f(k).
Một đa thức f(x) nhận x – k làm thừa số nếu và chỉ nếu f(k) = 0.
Nếu đa thức f ( x)  an x n  an 1 x n 1  ...  a1 x  a0 có các hệ số là số nguyên, khi đó nghiệm hữu tỷ
của đa thức f có dạng sau đây:
p
q
Trong đó p là ước của a0 và q là ước của an
Cho A và B là các số thực hoặc các biểu thức đại số. Nếu AB = 0 thì A = 0 hoặc B = 0
Một số biểu thức dạng tích đặc biệt:
-
Tổng và hiệu: (a  b)(a  b)  a 2  b2
Bình phương của nhị thức:
English for Mathematics
(a  b)2  a 2  2ab  b2
(a  b)2  a 2  2ab  b2
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Chapter 2. Algebra
(a  b)3  a3  3a 2b  3ab 2  b3
-
Lũy thừa bậc 3 của nhị thức:
-
Tổng của hai lũy thừa 3: a3  b3  (a  b)(a 2  ab  b2 )
-
Hiệu của hai lũy thừa 3: a3  b3  (a  b)(a 2  ab  b2 )
(a  b)3  a3  3a 2b  3ab 2  b3
4.2 Translate into Vietnamese:
4.2.1 Translate into the following sentences into Vietnamese:
a) An equation is a statement of the equality of two expressions.
b) A conditional equation, usually just called equation, is true only for certain values of the
symbols involved.
c) Symbols that can be assigned more than one value are called variables.
d) An algebraic expression is a collection of variables, constants and numbers connected
together by the arithmetic operations.
e) Expressions can be simplified by combining like terms and by factorizing similar terms.
f) A polynomial is an expression involving sums of powers of a variable.
g) Two polynomial are equal for all values of the variable if and only if their respective
coefficient are equal.
4.2.2 Translate into the following text into Vietnamese:
For the general quadratic equation ax 2  bx  c  0 , a  0 , the quantity b2  4ac in the solution
is called the discriminant, and the nature of the solution to the quadratic depends upon whether
this discriminant is positive, zero, or negative.
Positive discriminant: If the discriminant is a positive number then there are two distinct
solutions to the quadratic equation corresponding to the positive and negative values of the
square root of the discriminant. Graphically, this corresponds to the case where the graph of
y  ax 2  bx  c crosses the x-axis at two points.
Zero discriminant: If the discriminant is zero there is one real solution to the quadratic
equation. To maintain the pattern of description we say that there are two coincident solutions.
Graphically, this corresponds to the case where the graph of y  ax 2  bx  c touches the x-axis at
one point.
Negative discriminant: If the discriminant is negative the square root of the discriminant is an
imaginary number. Graphically, this corresponds to the case where the graph of y  ax 2  bx  c
neither crosses or touches the x-axis. In this case we have two solutions but they are not real
numbers. Each solution is a mixture of real and imaginary numbers. We call them complex
numbers.
5. EXERCISES:
5.1 Solve:
a) 4 x  3  5
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Chapter 2. Algebra
1
2
b) 2 ( x  2)  8
2
c) 5 x  x  8
d)
3x  1
12
 2
x2
( x  2)( x  1)
e)
3x  2
4

2
x  3x x  4
5.2 A simple pendulum consists of a compact mass suspended by a cord and performing small
oscillation about the vertical. If the length of the cord is L cm and the time taken to perform a
single complete swing is T s, the periodic time, the equation linking L and T is T  2
L
where
g
g  979.05cm.s-1 is the acceleration due to gravity. Calculate the period time T for pendulum with
lengths 50cm, 100 cm, and 200cm.
5.3 Rewrite the following expressions using the equations given:
a) x 2  y 2 where x =at and y  a 1  t 2
1
2
b) u 2  2as , where s  ut  at 2
5.4 Find the equation of the line through (1, 2) that is parallel to
a) y = 9x – 4
b) 3x + 7y = 5
- Draw these above straight lines.
- Find the equation of the straight line passes through (0, -5) and (3, 4). Tell whether the
lines are parallel, perpendicular, or neither.
5.5 Find the equation of the lines:
a) Line 1: through (1, 1) and (0,4).
Line 2: through (-2, 3) and (0, -3)
b) Line 1: through (-1, -1) and (2, 3)
Line 2: through (4, -3) and (-3, 2).
5.6 A triangle is formed by the straight lines x – y = 2, x + y = 3 and y = 2x – 1. What are the
coordinates of the vertices.
5.7 Solve mx 2  2 x  3  0 where m is an integer.
5.8 Work in pairs and solve the following problem:
a) A department of an office furniture manufacturing company makes two models of desk, the
standard model and the super model. Each item passes through two stages, cutting and
assembling. The total time available in any one week for cutting is 6000 minutes and for
assembly 10.000 minutes.
b) Each super desk requires 20 minutes of cutting and 30 minutes of assembling. Each
standard desk requires 10 minutes of cutting and 20 minutes of assembling. The profit is $120 on
each super desk and $70 on each standard desk. If x is the number of standard desks and y is the
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Chapter 2. Algebra
number of super desks manufactured in a week, find the production of each to give a maximum
profit of $38000 per week?
c) If 1  x  5 and 6  y  10 , what is the maximum value of
x
?
y
d) Factor f ( x)  2 x3  x 2  25x  12 given that f(4) = 0.
e) Find the rational zeros of f ( x)  x3  5x 2  2 x  8 .
5.9 Perform the following multiplication and simplify your result:
a) (5b + 2) (3b – 8)
b) (3x – 2) (2x2 + x – 5)
Puzzles: Four Adventurers and a Small Canoe
Four adventurers (Alex, Brook, Chris and Dusty) need to cross a river in a small canoe. The
canoe can only carry 100kg. Alex weighs 90kg, Brook weighs 80kg, Chris weighs 60kg and
Dusty weighs 40 kg, and they have 20kg of supplies. How do they get across?
Just for fun:
Student 1: Two fathers and two sons, how many people are there in the total?
Student 2: Too easy, four!
Student 1: Wrong
Student 2: Why is it wrong?
Student 1: I am the son of my father. My father is the son of my grandfather. So two fathers
and two sons are three people. Is that correct?
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Chapter 2. Algebra
Unit 2: FUNCTIONS AND THEIR GRAPHS.
__________________________________________
1. VOCABULARY AND GRAMMAR REVIEW:
1.1 Vocabulary:
- Amplitude (n): biên độ
- Asymmetrical (a): không đối xứng
- Asymptote (n): đường tiệm cận
- Axis (n): trục
- Composition (n) phần hợp
- Coordinate plane (n): hệ trục tọa độ
-
Crate (n): sọt đựng hoa quả
- Domain (n): miền (tập xác định)
- Even function (n): hàm chẵn
- Exponential function (n): hàm mũ
- Input (n): nguồn vào
- Logarithmic function (n): hàm logarit
- Quadrants (n): cung phần tư
- Odd function (n): hàm lẻ
- Output (n): nguồn ra
- Piecewise function (n): hàm số chia nhánh
- Putt (n) cú đánh bóng nhẹ vào lỗ đánh gôn
- Range (n) miền giá trị
- Relation (n): mối quan hệ
- Restricted domain (n): miền giới hạn
- Trigonometric functions (n): các hàm số lượng giác
- Vertex (n): đỉnh
1.2 Grammar review:
- Tenses
- Passive voice
- To infinitive
1.3 Practice exercises:
1.3.1 Fill in the blank with the correct words
Relation
Domain of relation
Range of relation
Function
Piecewise function
Coordinate plane
Exponential function
_______: a mapping or pairing of input values and output values.
_______: a relation where there is exactly one output for each input
_______: is a function represented by a combination equations each corresponding to a part
of the domain.
_______: set of output value.
_______: set of input value.
_______: Divided into 4 quadrants by the X-axis and Y-axis.
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Chapter 2. Algebra
_______: is a function that involves the expression b x where the base b is a positive other
than 1.
1.3.2 Fill in the blanks with the correct form of the word in the bracket:
TRANSFORMATION OF GRAPHS
In many cases the graphs of certain functions can be____(1)____(construct) by transforming
familiar graphs of other functions of a ____(2)___(simple) nature. For example, if we know the
graph of the function y = f (x), it is possible to graph functions of the form:
y  f ( x)   , y  f ( x   ) .
The graph of the function y = ax + b is ____(3)___(obtain) from the graph of the function y =
ax by shifting it |b| units (up or down, depending on the sign of b) along y – axis. It is also clear
that the ordinates of the graph of the function y  f ( x)   are obtained from the ordinates of the
graph of y = f (x) by ____(4)___(adjoin) the constant  . This ____(5)___(mean) we have to shift
the entire graph of y = f (x) parallel to the y – axis a total of |  | units upwards or downwards,
depending on the side of  .
If we want to use the graph of the function y = f (x) to ____(6)___(construction) the graph of
the function y  f ( x   ) , then we have to translate the graph of the function y = f (x) a total of
|  | scale units to the left if   0 and to the right if   0 .
1.3.3 Choose the suitable word in the table to fill in the blanks:
Where
Substituting Function
With
Into
Of
Single
More
Every
Consist
Evaluating
Who
Is
Accordingly According
On
One
Very
How
numbers
Much
We are already familiar___(1)___ algebraic expressions; they___(2)___of symbols in the
form of numbers, variables and constants connected together by arithmetic operations. We are
also familiar with the process of_____(3)____expression by___(4)___symbols with____(5)__. If
the process of evaluating an algebraic expression produces a____(6)___ number then the process
of evaluation is a____(7)___. For example, the expression
x 2 ( x  1)
has a single value for every
x2  x  1
value of x substituted into the expression.____(8)____, we can define the function
f____(9)___ f : f ( x) 
x 2 ( x  1)
x2  x  1
1.3.4 Put the word in the correct order to make the meaningful sentences:
a) The / and / every / number / in/ range / of / a / function / consists / of / all / those / numbers
f(x) / that / correspond / to / each / the / domain.
b) A / function / to /produce /an / output / processes/ an / input /number / number.
c) A /one-to-one / a / unique / domain / value / function / is / such / any / given / that / for /
range / value / there/ corresponds.
d) If / a / function f / is / the /function / is / said/ such /that / f(-x) = -f(x) ,/ to / be / odd
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Chapter 2. Algebra
e) A / many-to–one / function / is / than / one / corresponding / domain / such / that/ for / any
/ given / range / value / there / may / be / more /value.
f) The / about / the / vertical / coordinate / axis / graph / of / an / even / function / is /
symmetrical.
2. READING:
2.1 FUNCTION:
A system consists of an input, a process that acts on the input and output that is the result of
the processing. The processing unit of the system converts the input into the output. This notion
of a system is used to define a mathematical function. If the input and output of a system are
numbers then provided that a system are numbers then provided that a single input number is
processed into only one output number the processor is called a function.
If the input number is labeled as x and the function labeled as f then the output is the effect of
f acting on x and is accordingly labeled f(x) – read as “f of x”.
All the numbers that a function f can process are collectively called the domain of a function.
Sometimes, we may wish to restrict the domain to a smaller collection of numbers than the
totality of the number that it can process. This smaller collection of numbers is called a
restricted domain.
The range of a function consists of all those numbers f(x) that correspond to each and every
number in the domain. For example, where f : f ( x)  x3 has the domain and a range that both
consists of all the real numbers. We represent this domain by   x   . While the function f,
where f : f ( x) 
1
has a domain and a range that both consider of all the real numbers excluding
x
0. The domain is written as   x  0 and 0  x   .
Functions can be subjected to the ordinary rules of arithmetic provided care is taken to define
their common domain. Let f and g be any two functions. A new function h can be defined by
performing any of the four basic operations (addition, subtraction, multiplication and division) on
f and g.
- Addition: h(x) = f(x) + g(x)
- Subtraction: h(x) = f(x) – g(x)
- Multiplication: h(x) = f(x)g(x)
-
Division: h( x) 
f ( x)
g ( x)
The domain of h consists of the x-values that are in the domain of f and g. Besides, the
domain of a quotient does not include x values for which the denominator is zero.
The composition of the function f with the function g is h(x) = f(g(x)). The domain of h is the
set of all x – values such that x is in the domain of g and g(x) is in the domain of f.
One-to-one function is such that for any given range value their correspondences a unique
domain value. However, a many-to-one function is such that for any given range value there may
be more than one corresponding domain value, for example f ( x)  x2 .
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Chapter 2. Algebra
The zero (or root) of a function f (x) is that value of the argument x for which the function
vanishes. Graphically, the zeros of a function are the points of intersection of the graph with the
x-axis. If a function f is such that f(-x) = f(x) the function is said to be even. The graph of an
even function is symmetrical about the vertical coordinate axis. If a function f is such that f(-x) =
-f(x) , the function is said to be odd, the graph of odd function is symmetric about the origin.
Not all functions are either odd or even. However, it may be possible to create an even and
odd function from one that is neither – we call them the even and odd parts of the function
respectively. Given the function f then if both f(x) and f(-x) are defined:
f ( x)  f ( x)
defines an even function called the even part of f(x).
2
f ( x)  f ( x)
defines an odd function called the odd part of f(x).
f 0 ( x) 
2
f e ( x) 
Comprehension check:
Answer the following questions:
1) What is a function? Give two examples of one-to-one functions and many-to-one
functions?
2) Can we multiply or divide two arbitrary functions? Why and Why not?
3) What are the domain and the range of a function? Give some examples of function with
their domains and their ranges?
4) Let f ( x)  x2 with domain 0  x  4 and g ( x)  3x with domain 2  x  6 . What are the
domain of h = f – g; k = f + g.
5) What are the characteristics of an even function and an odd function? Give examples of
even function, odd function?
6) Can we create an even and odd function? How?
7) Which of the following are even, which are odd and which are neither?
a) f ( x)  x / 4
b) f ( x)  x1/3 c) f ( x)   x3  x
d) f ( x)  x4
2.2 THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS:
Any function f with the output of the form f ( x)  a x is called an exponential function, where
a is referred to as the base of the function.
If a > 1 then the output is always positive. The curve crosses the vertical axis at f(0) = 1, and
as x increases the exponential curve also increases without bound. As x becomes increasingly
negative the curve approaches the x-axis. However, the curve never actually meets the x-axis.
If a < 1 the graph of f ( x)  a x is a mirror image of the graph with a > 1, where the mirror is
taken to lie along the vertical axis.
Logarithm with base b: Let b and y be positive numbers, b  1. The logarithm of y with base b
is denoted by logb y and is defined as follows: logb y  x , if and only if b x  y . The expression
logb y is read as “log base b of y”.
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Chapter 2. Algebra
The logarithmic function is defined by f ( x)  log a x . The logarithmic function is not defined
for negative values of x and that as x increases so the logarithm increases without bound. The
logarithmic passes through the x-axis at the point x = 1, which is a pictorial display of the
statement that: log a 1  0 or alternatively, a 0  1 . As x approaches zero so the curve approaches
the vertical axis, the output becoming increasingly negative. The natural logarithm has as base
the irrational exponential number e and is denoted as ln(x). The shape of its graph is similar to
that of the exponential function but with a different orientation.
The graph of y  logb ( x  h)  k has these characteristics:
-
The line x = h is a vertical asymptote.
The domain is (h, ) , and the range is (, )
Comprehension check
Answer the following question:
1) What is an exponential function? Give an example?
2) What is the characteristic of the graph of an exponential function?
3) What is a logarithmic function? Give some example?
4) What is the natural logarithm?
5) What is the characteristic of the graph of an logarithmic function?
3. SPEAKING – WRITING – LISTENING - DISCUSSION
3.1 Discussion:
a) What is the characteristic of the function increasing in an interval lying in the domain of
definition? How’s about the decreasing function? Give examples?
b) Give some examples of trigonometric functions you have learned? What are their domains
and their ranges?
c) Graph these above trigonometric functions. What are the characteristic of their graphs?
d) What is periodic function? Give some examples of periodic functions, and the period of
these functions?
3.2 Writing:
Write a short paragraph (300 words) about the inverse of a function or about the piecewise
function.
3.3 Listening:
Listen to the tape and fill in the blanks
A____(1)___ processes an input____(2)___ to produce an output number.___(3)___a number
can be input into a function that the function____(4)___process. For example, if
a____(5)___displays the number 0 as an input then_____(6)__the 1/x function key produces
an____(7)___display. The function that produces the_____(8)___ of a number cannot process the
number 0 because we have not defined what is meant by division by____(9)___. Clearly, we
need to____(10)___ which numbers a function can process and which numbers it cannot.
All the numbers that a function f can process are_____(11)___ called the____(12)___of the
function.
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Chapter 2. Algebra
4. TRANSLATION
4.1.1 Translate each of the following sentences into Vietnamese:
a) Two functions f and g can be added or subtracted to form their sum or difference f  g .
b) Functions can be subjected to the ordinary rules of arithmetic provided care is taken to
define their common domain.
c) The graph of a one-to-one function is such that any vertical line through a domain value
and any horizontal line through a range value will intersect with the graph once at most.
d) The amplitude of a periodic function is defined as the difference between the average value
and the maximum value of the output taken over a period.
4.1 .2 Translate the following text into Vietnamese
The graph of y = a|x-h| + h has the following characteristics:
- The graph has vertex (h, k), and is symmetric in the line x = h.
- The graph is V-shaped. It opens up if a > 0 and open down if a<0.
- The graph is wider than the graph of y = |x| if |a| s<1.
- The graph is narrower than the graph of y = |x| if |a| >1.
The graph of y  a( x  h)2  k is a parabola with the vertex (h, k) and its axis of symmetry is x
= h. The graph of y  a( x  p)( x  q) is the parabola. The x-intercept are p and q. Axis of
symmetry is halfway between (p, 0) and (q, 0).
4.2 Translate into English
Đồ thị của hàm số y  ax 2  bx  c là một parabol với các tính chất sau:
- Parabol mở ra (open up) nếu a > 0 và úp xuống (open down) nếu a< 0. Parabol sẽ rộng hơn
đồ thị của hàm y  x 2 nếu |a| <1 và hẹp hơn đồ thị của hàm y  x 2 nếu |a| >1. Hoành độ của đỉnh
parabol là
b
b
và trục đối xứng là đường thẳng đứng x 
.
2a
2a
5. PRACTICE EXERCISES:
5.1 Let f(x) = 2x and g ( x)  x1/2 .
a)
Find the product of the functions, the quotient of the functions and the domains of the
product and the quotient.
b)
Evaluate f(x) when (a) x = 3 and (b) x = -2, where
 x  4, if x  1
f ( x)  
3x  1, if x  1
c) Graph the above piecewise function.
5.2 Shipping costs $3 on purchases up to $10, $5 on purchases up to $50, and $8 on purchases
over $50 up to $100. Write the piecewise function for this situation, and graph it.
5.3 Graph each of the following functions. Identify the x-intercepts and the points where the
local maximums and local minimums occur.
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Chapter 2. Algebra
a ) f ( x)   x 3  4 x 2  x  4
b) f ( x)   x 4  2 x3  2 x 2  3x
5.4 The roof of a building can be modeled by the function y 
4
| x  9 | 12 , where x and y
3
are measured in feet and the x-axis represents the base of the roof.
a) Graph the function
b) Interpret the domain and the range of the function.
5.5 While playing miniature golf, you hit the ball off of a wall to try to make it in the hole.
The ball is located at (1, 2) and the hole is at (6, 2). The point where the ball hits the wall is (3, 0)
a) Write an equation for the path of the ball
b) Do you make your putt?
5.6 The path of a ball after you kicked it can be modeled by y  0.1x 2  4 x , where x is the
horizontal distance in feet and y is the height in feet of the ball. What was the height of the ball at
its highest point? How many feet had the ball traveled horizontally at this height.
5.7 The table shows the path taken by a paper airplane where x measure horizontal distance
(meters) and y measure height (meters). Find a quadratic model in standard form for the data.
Horizo
0
1
2
3
4
5
6
7
ntal
distance, x
Height
0.7
1.75
2.4
2.7
2.8
2.3
1.6
0.4
,y
5.8 A rectangular swimming pool has a volume of 512 cubic feet. The pool’s dimension are x
feet deep by 6x – 8 feet long by 6x – 16 feet wide. How deep is the pool?
5.9 A woman goes to the bank to get change for $20.00 and receive equal number each of
quarter, dimes and nickels. How many coins did she receive total?
5.10
If x 2  y 2  15 and xy = 5, What is the value of x+y?
5.11 Ed and Lori go shopping. Ed spends $30 more than Lori in the first store and Lori spends
$12 less than Ed in the second store. Which of the following must be true about Lori’s total
spending in the two stores compares to Ed’s?
A) Lori spent
2
of what Ed spent.
5
B) Lori spent $18 less than Ed.
C) Lori spent $21 less than Ed
D) Lori spent $42 less than Ed
E) Lori spent $42 more than Ed.
5.12 Thad bikes m miles every h hours. At this rate, how many miles would he bikes in 45
minutes?
5.13 If 3 is subtracted from a certain number, the result is 5 less than twice the number what
is the number?
5.14
If 10x + y = 8 and 7x – y = 9, what is the value of 3x + 2y?
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Chapter 2. Algebra
5.15 Pat has s grams of strawberries and uses 40 percent of the strawberries to make pies,
each of which requires p grams. The rest of the strawberries are used to make pints of jam, each
of which requires j grams. What is the number of pints of jam Pat can make?
5.16 Evaluate: log 2 64 ; log1/3 27 .
5.17 Graph these following functions:
a) f ( x)  log 2 ( x  3)  7 b)
b) f ( x)  2x
1
c) f ( x)   
2
x
d) f ( x)  ln( x)
6. READING FOR REFERENCE:
MAPPING
Now, we shall concern ourselves with another of the important concepts of mathematics - the
notion of mapping. But first let us try an experiment designed to yield some information about
our mental habits. Visualize your best friend. Of course, the image of a certain individual forms
in your mind. But did you notice that accompanying this image is a name – the name of your
friend? Not only did “see” your friend, but you also thought of his name. In fact, is it possible for
you to visualize any individual without his name immediately emerging in your memory? Try!
Furthermore, is it possible for you to think of the name of an individual, at the same time,
visualizing that individual? The point of the proceeding experiment is to demonstrate that we
habitually link together a person which his name, we seldom think of one without the other. Let
us see what there is of mathematical value in the above observation. First, let us state the
essentials of the situation. On the one hand, we have a set of persons, on the other hand, the set of
names of these persons.
With each member of the first set we associate, in a natural way a member of the second set.
It is in the process of associating members of one set with the members of another set that
something new has been created. Let us analyse the situation mathematically. Denote the set of
persons by “P” and the set of names by “N”. We want to associate with each member of “P” an
appropriately chosen member of N; in fact we want to create a mathematical object which will
characterize this association of members of N with members P. We rely on one simple
observation; there is no better way of indicating that two objects are link together than by
actually writing down the names of objects, one after the other, i.e., we indicate that two object
are associated by pairing the objects. Now we see the importance of ordered pairs. The ordered
pair (a, b) can be used to indicate that a and b are linked together.
Now we know how to characterize associating members of N with members of P, construct
the subset P  N obtained by pairing with each person his name. The resulting set of ordered pairs
express mathematically the associating process described above, since the person and the name
that belong together appear in the same ordered pair.
Now a definition. Let “A” and “B” denote any non empty set. A subset of A  B , say  is said
to be a mapping of A into B if each member of A is a first term of exactly one ordered pair in  .
Moreover, we shall say that that the mapping  associates with a given member of A, say a, the
member of B paired with a. Thus, if (a, b)   we shall say that “b” is associated with “a” under
the mapping  , b is also called the image of a under the mapping. Note that the subset
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Chapter 2. Algebra
P  N constructed above is a mapping of P into N. Thus our notion of a mapping of A into N
permits us to characterize mathematically the intuitive idea of associating a member of B with
each member of A.
Under the intuitive idea, b is associated with a; this is represented by the mathematical
assertion (a, b)   . In short, the set  characterize the intuitive idea of associating a member of
B with a member of A. If  is a mapping of A into B such that each member of B is a second
term of at least one member of  , then we shall say that  is a mapping of A into B.
Furthermore, if a mapping of A into B such that no member of B is a second term of two ordered
pairs in the mapping, then we say that this subset of A  B is one to one mapping of A into B. For
example: {(1,3); (2,4); (3,5); (4,6)} is one to one mapping of {1,2,3,4} into{3,4,5,6,7}. If  is
both a one to one mapping of A into B and a mapping of A onto B, then  is said to be a one to
one mapping of A onto B.
Comprehension check:
Answer the following questions:
1. Which experiment in this text is designed to yield some information about our mental
habit?
2. Does the image of a man usually accompany his name?
3. Does one necessarily visualize a man when hearing his name?
4. Why do we habitually link together a person and his name?
5. What do we show by pairing objects?
6. How do we characterize associating members of N with the member of B?
7. When do we say that  is a mapping of A into B?
8. Under what condition is a subset of A x B said to be a mapping of A into B?
9. What do we mean by saying that the subset of A x B is one to one mapping of A into B?
Puzzles: Crates of Fruit
You are on an island and there are three crates of fruit that have washed up in front of you.
One crate contains only apples. One crate contains only oranges. The other crate contains both
apples and oranges.
Each crate is labeled. One reads "apples", one reads "oranges", and one reads "apples and
oranges". You know that NONE of the crates have been labeled correctly - they are all wrong.
If you can only take out and look at just one of the pieces of fruit from just one of the crates, how
can you label ALL of the crates correctly?
Just for fun: One and Zero
The ONE stood itself up tall and thin as a rod, and imagining itself a flagpole, swaggered
from town to town, gathering up simple minded and empty ZEROS. “Follow me! We’ll be
invincible! A few friends will make us thousands, and with a few more we’ll be millions!”
The mathematician watched the procession.
“Wonderful!” he laughed. “A few buglers make a big shot.”
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Chapter 2. Algebra
Unit 3: MATRIX –DETERMINANT OF A MATRIX
AND LINEAR SYSTEM
________________________________________________________
1. VOCABULARY AND GRAMMAR REVIEW:
1.1 Vocabulary:
- Algebraic invariants (n): lượng đại số bất biến
- Astronomy (n): thiên văn học
- Column vector (n): Vector cột
- Determinant (n): định thức
- Diagonal (n): đường chéo
- Diagonal matrix (n): ma trận chéo
- Differential geometry (n): hình học vi phân
- Estimate (v): ước lượng
- Geneticist (n) nhà di truyền học
- Homogenous system (n): hệ phương trình thuần nhất
- Implicit (a): ẩn
- Inconsistent (a): mâu thuẫn
- Invariants (n): lượng bất biến
- Indeterminate (a): vô định
- Linear equations (n): phương trình tuyến tính
- Linear transformations (n): phép biến đổi tuyến tính
- Lower triangular matrix (n): ma trận tam giác dưới
- Magnetism (n): hiện tượng từ tính
- Matrix (n): ma trận
- Nonsingular matrix (n): ma trận khả nghịch
- Nontrivial solution (n): nghiệm không tầm thường
- Oil refinery (n): lọc dầu
- Prodigy (n): người phi thường
- Quantum mechanics (n): cơ học lượng tử
- Row vector (n): Vector dòng
- Safari (n): cuộc đi săn
- Scalar (n): sự vô hướng
- Singular matrix (n): ma trận không khả nghịch
- Skew symmetric matrix (n): ma trận phản đối xứng
- Square matrix (n): ma trận vuông
- Statistician (n): nhà thống kê
- Spectrum (n): phổ
- Transpose (v): chuyển vị
- Trivial solution (n): nghiệm tầm thường
- Unit matrix (n): ma trận đơn vị
- Upper triangular matrix (n): ma trận tam giác trên.
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Chapter 2. Algebra
1.2 Grammar review:
- Tense
- Verb form
1.3 Practice Exercises:
1.3.1 Choose the correct words to fill in the blanks:
Transpose
Diagonal
Upper triangular
Skew symmetric
Lower triangular
Square Scalar
Nonsingular
Unit
singular
A matrix is an array of numbers arranged in regular rows and columns. For example,
1 2 
1 2 3
A
or B  

.
4 5 6
3 6 
The first matrix is referred to as 2 x 2 matrix and the second as a 2 x 3 matrix. If a matrix has
i rows and j columns, it is called an i by j (i x j) matrix .
There are many types of matrix, some of which are listed below:
- ____(1)____matrix: A ___(2)___ matrix is a matrix with the same number of row as
columns.
- _____(3)___ matrix: A ___(4)____ matrix is a square matrix with zeros everywhere
except down the leading diagonal.
- ____(5)____ matrix: is a diagonal matrix whose diagonal elements are equal.
- ____(6)____ matrix and _____(7)___matrix: An n x n matrix A  [aij ] is called
_____(8)____ if aij  0 for i > j. If aij  0 for i < j is called _____(9)_____. A ___(10)____ matrix
is both upper triangular and lower triangular.
- _____(11)____ of a matrix: The _______(12)____ AT of matrix A is the matrix formed
from A by interchanging rows for column.
- _____(13)____ matrix: A symmetric matrix is a matrix whose (i, j) element is the same as
the (j, i) element. The (i, j) element is the element in the ith row and jth column. Moreover, a
matrix A with real entries is called symmetric if AT  A .
- ____(8)_____ matrix: A matrix A with real entries is called _____(8)___ if AT   A .
- ____(9)____ – ____(10)___ matrix: An nxn matrix A is called _____(9)___, or invertible
if there exists an nxn matrix B such that AB = BA = In, such a B is called an inverse of A.
Otherwise, A is called____(10)____ or noninvertible matrix.
- ____(11)____ matrix: A unit matrix is a ___(11)___ matrix with unity down the leading
diagonal.
- Row vector: A matrix that consists of a single row is called a row vector.
- Column vector: A matrix that consists of a single column is called a column vector.
1.3.2 Fill in the blanks with the correct form of the words in the bracket.
THE ARITHMETIC OF MATRICES
Matrices can be (add) ____(1)___, subtracted, multiplied by a scalar and multiplied together
provide they satisfy certain criteria.
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Chapter 2. Algebra
Addition and Subtraction: To add two matrices together they must both have the same
number of rows as well as columns. Their sum ____(2)___(be) then obtained by adding
corresponding elements to form a matrix with the same number of ____(3)___ (row) and
columns.
Multiplication by a scalar: To multiply a matrix by a single number each element of the
matrix is ____(4)____(multiply) by that number.
Multiplication two matrices: Two matrices can be multiplied together ____(5)___ (provide)
that the matrix on the left has as many columns as the matrix on the right has rows. The principle
is that each number in the nth row of the left-hand matrix is multiplied by each corresponding
term of the mth column of the right-hand matrix. These products ____(6)___(be) then added to
form the element in the nth row, mth column of the product matrix. As a consequence, the product
of matrices PQ where P is an n x r matrix and Q is an r x m matrix is an n x m matrix.
Multiplication by a Unit matrix: If A is an i x j matrix and I is the j x j unit matrix then AI =
A.
Inverting Matrices: Division of matrices cannot be defined by an operation similar in effect of
division is that of multiplication of a matrix by its inverse. If A and B are two
____(7)___(inverting) matrices such that AB = I, where I is a unit matrix, then matrix B is called
the inverse matrix of A and is written as A1 , matrix A is also called nonsingular or
____(8)____(inversely). We could also claim that A was the inverse of matrix B and write A as
B1 . More correctly, B is the right-hand inverse of A and A is the left-hand inverse of B. The
left-hand and right-hand inverses of the 2 x 2 matrices are equal.
1.3.3 Match column A with column B to make the meaningful sentences:
A
B
1. The inverse of an invertible
matrix
2. A square matrix over
invertible
is
3. If A is an invertible matrix,
a. Is unique
b. If and only if it is a product of
elementary matrices.
c. Its unique inverse is denoted by
A .
1
4. If A and B are invertible matrices
of the same order,
d. Then
1
1
AB
is
invertible
and
1
( AB)  B A
2. READING:
2.1 MATRICES
Although the idea of a matrix was implicit in the quaternion (4-tuples) of N. Hamilton and
also in the “extended magnitude” (n–tuples) of H. Grassmann, the credit for inventing matrices is
usually given to Cayley with a date of 1857, even though Hamilton obtained one of two isolated
results in 1852. Cayley said that he got the idea of a matrix “either directly from that of a
determinant, or as a convenient mode of expression of the equations x = ax + by, y’= cx + dy’’.
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Chapter 2. Algebra
He represented this transformation and developed an algebra of matrices by observing properties
of transformation on linear equations;
 x '  ax  by  a b 



 y '  cx  dy  c d 
Cayley also showed that a quaternion could be represented in matrix form as shown above
where a, b, c, d are suitable complex numbers. For example, if we let the quaternion units 1, i, j, k
be represented by
1 0   i 0   0 1 
0 i 
0 1  , 0 i  ,  1 0 and  i 0

 
 



the quaternion 4  5i  6 j  7k can be written as shown below:
 4  5i 6  7i 
 6  7i 4  5i 


This led P.G Tait, a disciple of Hamilton, to conclude erroneously that Cayley had used
quaternion as his motivation for matrices. It was shown by Hamilton in his theory of quaternion
that one could have a logical system in which the multiplication also is non-commutative. In
1925 Heisenberg discovered that the algebra of matrices is just right for the non commutative
maths describing phenomena in quantum mechanics.
Cayley’s theory of matrices grew out of his interest in linear transformations and algebraic
invariants, an interest be shared with J.J Silvester. In collaboration with J.J Silvester, Cayley
began the work on the theory of algebraic invariants which had been in the air for some time and
which, like matrices, received some of its motivation form determinants. They investigated
algebraic expressions that remained invariants (unchanged except, possibly, for a constant factor)
when the variables were transformed by substitutions representing translations, rotations,
dilatations, (“stretching” from the origin), reflections about an axis, and so forth.
There are three fundamental operations in matrix algebra; addition multiplication and
transposition, the last not occurring in ordinary algebra. The law of multiplication of matrices
which Cayley invented and his successors have approved takes its rise in the theory of linear
transformation. Linear combinations of matrices with scalar coefficient obey the rules of ordinary
algebra. A transformation is a permutation which interchanges two numbers and leaves the other
fixed, or in other word: the formal operation leading from x to x’ and also that leading from x to
x’ and also that leading from x’ to x is called transposition. A matrix of m rows and n columns
has rank r, when not all its minor determinants of order r vanish, while of order r +1 do. A marix
and its transposition have the same rank. The rank of a square matrix is the greatest number of its
rows or columns which are linearly independent.
Today, matrix theory is usually considered as the main subject of linear algebra and it is a
mathematical tool of the social scientist, geneticist, statistician, engineer and physical scientist.
Comprehension check:
A. Are the statements true (T) or false (F)? Correct the false statements
a) _________ Cayley was the first inventor of matrices?
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Chapter 2. Algebra
b) _________Cayley’s idea of a matrix comes from Hamilton’s theory of quaternion.
c) _________Properties of transformation on linear equations serve as the basis of Cayley’s
theory of matrices.
d) _________The law of multiplication of matrices does not relate to the theory of linear
transformation.
B. Answer the following questions
a) Who was the first to create a matrix? When was this?
b) Did anyone obtain the idea of a matrix before him?
c) What did Halmiton’s theory of quaternion help Cayley to work his out matrix theory?
d) What did Heisenberg discover in 1925?
e) Who did Cayley collaborate with on the theory of algebraic invariants? What did they
investigate?
f) What are three fundamental operations in matrix algebra?
2.2 LINEAR SYSTEM
One of the most frequently recurring practical problems in many fields of study such as
mathematics, physics, biology, chemistry, economics, all phases of engineer, operation research,
and the social sciences – is that of solving a system of linear equations. The equation
a1 x1  a2 x2  ...  an xn  b (1), which expresses the real or complex quantity b in terms of the
unknowns x1 , x2 ,..., xn and the real or complex constants a1 , a2 ,..., an is called a linear equation.
More generally, a system of m linear equations in n unknowns x1 , x2 ,..., xn or a linear system is a
set of m linear equations each in n unknowns. A linear system can conveniently be written as
 a11 x1  a12 x2  ...  a1n xn  b1
 a x  a x  ...  a x  b
 21 1 22 2
2n n
2
(2)


 am1 x1  am 2 x2  ...  amn xn  bm
If the linear system (2) has no solution, it is said to be inconsistent, if it has a solution, it is
called consistent. If b1  b2  ...  bm  0 , then (2) is called the homogenous system. Note that,
x1  x2  ...  xn  0 is always a solution to a homogenous system, it is called the trivial solution. A
solution to homogenous system in which not all of x1 , x2 ,..., xn are zero is called a nontrivial
solution. Two linear systems are said to be equivalent if they both have exactly the same
solutions. To find solutions to a linear system, we shall use a technique called the method of
elimination; that is, we eliminate some variables by adding a multiple of one equation to another
equation.
Comprehension check:
A) Choose true (T) or false (F):
1) A linear system is a system of m linear equations in n unknowns.
2) A linear system has at least one solution is said to be consistent.
3) A linear system can have infinite solutions.
4) A linear system having no solution is said to be inconsistent.
5) A homogenous system is a linear system if it has a trivial solution.
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Chapter 2. Algebra
6) A linear system always has the trivial solution.
7) To solve a linear system we can add a multiple of one equation to the others in order to
eliminate some variables.
3. SPEAKING – LISTENING – WRITING – DISCUSSION:
3.1 Discussion: Work in group to answer the following question
1) What is a matrix? Give some examples of 5 x 6 matrix and 4 x 4 matrix?
2) What is the characteristic of square matrix? Symmetric matrix? Skew matrix? Give
example for each type of those matrices?
3) What is the characteristic of singular matrix? Give example of singular matrix and
nonsingular matrix?
4) How can we transpose a matrix? Give example?
5) Give example of scalar matrix?
6) What are the differences between the upper triangular matrix and lower triangular matrix?
Give example?
7) Besides these above matrices, could you give some more different types of matrices and
their examples?
8) What is the method of finding the inverse matrix of a square matrix A?
9) What is the rank of matrix A? How to find rank of a matrix?
10) Describe some methods of solving linear system you have ever learned. Give some
examples.
11) Work in group to prove or disprove these following statements
a) det (A + B) = det A + det B and det( A1B) 
det( B)
.
det( A)
b) If det (A) = 0 then A has at least two equal rows.
c) A is singular if and only if det (A) = 0.
12) Work in groups to prove these following theorems:
a) Let A be an nxn matrix and let the homogenous system Ax = 0 have only the trivial
solution, then A is row equivalent to In.
b) A is nonsingular if and only if A is a product of elementary matrices.
c) A is nonsingular if and only if A is row equivalent to In.
1 1 0 
d) Find all values of a for which the inverse of A  1 0 0 
1 2 a 
e) Let A, B and C be 2x2 matrices with det(A) = 3, det (B) = -2, and det (C) = 4. Compute
det(6 AT BC 1 )
f) Prove or disprove: For 3x3 matrix A, if B is the matrix obtained by adding 5 to each entry
of A, then det (B) = 5 + det (A).
g) Let A be a lower triangular matrix. Prove that A is singular if and only if some diagonal
entry of A is zero.
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Chapter 2. Algebra
3.2 Writing:
Write a short paragraph (more than 300 words) about the application of the determinant of a
matrix.
3.3 Listening:
Listen to the tape and fill in the blanks:
There____(1)___ various approaches to that____(2)__ of mathematics known as
_____(3)____algebra. Different approaches ____(4)___different_____(5)___ of the subject such
as matrices, ______(6)_____, or computational methods. Linear algebra is in essence a
____(7)___ of vector spaces, and this study of vector space is primarily_____(8)___ to finite –
____(9)____vector spaces. The real coordinate spaces, in addition to being ____(10)____ in
many applications, furnish excellent intuitive models of abstract finite-dimensional
vector____(11)___.
4. TRANSLATION:
4.1 Translate into Vietnamese:
4.1.1 Translate the following sentences into Vietnamese:
a) An nxn matrix A is invertible if and only if det(A) ≠ 0.
b) For any square matrix A over F, the polynomial det(A – xI) = 0 is the characteristic
polynomial of A. The equation det(A – xI) = 0 is the characteristic equation of A, and the
solutions of det (A – xI) = 0 are called the eigenvalues of A. Th set of all eigenvalues of A is
called the spectrum of A.
c) Let  be an eigenvalue of the nxn matrix A. Then an eigenvector of A associated with  is
a nonzero nx1 matrix X such that AX   X .
d) If A is invertible then AT is invertible and ( AT )1   A1 
T
4.1.2 Translate the following text into Vietnamese:
A) Translate into Vietnamese:
Carl Friedrich Gauss (1777 – 1855) was born into a poor working-class family in Brunswick,
Germany, and died in Gottingen, Germany, the most famous mathematician in the world. He was
a child prodigy with a genius that did not impress his father. However, his teachers were
impressed enough to arrange for the Duke of Brunswick to provide a scholarship for Gauss at the
local secondary school. As a teenager there, he made original discoveries in number theory and
began to speculate about non-Euclidean geometry. His scientific publications include important
contributions in number theory, mathematical astronomy, mathematical geography, statistics,
differential geometry and magnetism. In his research Gauss used a method of calculation that
later generations generalized to row reduction of matrices and named in his honor, although the
method was used in China almost 2000 years earlier.
4.2 Translate into English:
4.2.1 Phép nhân hai ma trận:
Chúng ta có thể khái quát vài nét khác nhau giữa việc nhân hai ma trận và nhân hai số thực
như sau:
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Chapter 2. Algebra
1. AB có thể không bằng BA.
2. AB có thể là ma trận không với A  0 và B  0
AB có thể bằng AC với B  C
A.
4.2.2 Các tính chất của định thức :
a) Nếu A là ma trận thì det( A)  det( AT )
b) Nếu B là ma trận rút ra từ ma trận A bằng cách hoán đổi hai dòng (hai cột ) khác nhau của
ma trận A thì det(B) = - det (A).
c) Nếu hai dòng (cột) của ma trận A bằng nhau, thì det (A) = 0.
d) Nếu ma trận A có một dòng (cột) bằng không thì det (A) = 0.
e) Nếu ma trận B được rút ra từ ma trận A bằng cách nhân một dòng (cột) của A với một số
thực k thì det (B) = k det (A).
f) Nếu A là ma trận vuông n x n, thì A là ma trận khả nghịch nếu và chỉ nếu det A  0 .
5. EXERCISES:
3 3 1
5.1 Find AAT where A  5 0 4 
0 8 6 
6 3
0 2
6
,
B

,
C

6

2
,
D

5.2 Given matrices: A  



 3 1
 4
1 4 


 
Calculate each of the following:
a) 6A + B
b) AB
c) AD
d) CB
5.3 Perform the indicated operation, if possible:
 2  2 0 7 
 2 2 3 2
a)    
b) 2 



1  1 1 0 
0 1  1 0
1 4 0 
 1 5 3 3  8 9 
 2 3 
c) 
1 2 1  d)  




1 3 3 0  7 6
 1 5 


0 1 4
5.4 Tell whether the matrices are equal or not equal.
3 
24



a)  2 6
8 and  2
0.375 8
8
4




 1 3 
1 3
and
b) 

1 2 
 1 2


5.5 Solve the matrix operation for x and y:
  2 3  y 7    10 20


5  4 10   10 30 
a) 2  
 x
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Chapter 2. Algebra
 3
x  y
2 
0 2
b)   

  

  5 0   1 3  6 3
5.6 Use matrix to organize the following information
The population estimate for Gallatin County for July 1, 2001 was 36,315 males and 33,532
females. For Glacier County the estimate was 6452 males and 6633 females.
The population estimate for Gallatin County for July 1, 2002 was 36,998 males and 34,208
females. For Glacier County the estimate was 6434 males and 6672 females.
a b 
5.7 Find the inverse matrix of A  
 where a, b, c, d are not equal to zero.
c d 
5.8 If AB = BA and p is a nonnegative integer, show that ( AB) p  A p B p .
5.9 If p is a nonnegative integer and c is a scalar, show that (cA) p  c p A p .
5.10 Tim wants 4 notebooks, 5 pens and 10 pencils for the new school year. Cathy wants 5
notebooks, 8 pens and 8 pencils. Each notebook is $4, each pen is $2 and each pencil is $1. Use
matrix multiplication to find the total cost for each person.
5.11 A diet research project consists of adults and children both sexes. This composition of
the participants in the project is given by the matrix:
Adults Children
 80 120  Male
A

100 200 Female
The number of daily grams of protein, fat and carbohydrate consumed by each child and
adults is given by the matrix
Protein Fat
 20
B
10
Carbohydrate
20
20
20  Adult
30  Child
a) How many grams of protein are consumed daily by the males in the project?
b) How many grams of fat are consumed daily by the females in the project?
5.12 The director of a trust fund has $100,000 to invest. The rules of the trust state that both a
certificate of deposit (CD) and a long-term bond must be used. The director’s goal is to have the
trust yield $7800 on its investments for the year. The CD chosen returns 5% per annum and the
bond 9%. Find out the amount of money the trust should invest in the CD and in the long-term
bond to satisfy the rules.
5.13 A manufacturer makes three types of chemical products: A, B and C. Each product must
go through two processing machines: X and Y. The products require the following times in
machines X and Y:
- One ton of A requires 2 hours in machine X and 2 hours in machine Y.
- One ton of B requires 3 hours in machine X and 2 hours in machine Y.
- One ton of C requires 4 hours in machine X and 3 hours in machine Y.
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Chapter 2. Algebra
Machine X is available 80 hours per week and machine Y is available 60 hours per week.
Since management does not want to keep the expensive machines X and Y idle, it would like to
know how many tons of each product to make so that the machines are fully utilized. It is
assumed that the manufacturer can sell as much pf the product as is made.
5.14 An oil refinery produces low-sulfur and high-sulfur fuel. Each ton of low-sulfur fuel
requires 5 minutes in the blending plant and 4 minutes in the refining plant; each ton of highsulfur fuel requires 4 minutes in the blending plant and 2 minutes in the refining plant. If the
blending plant is available for 3 hours and the refining plant is available for 2 hours, how many
tons of each type of fuel should be manufactured so that the plants are fully used?
5.15 A plastic manufacturer makes two types of plastic: regular and special. Each ton of
regular plastic requires 2 hours in plant A and 5 hours in plant B; each ton of special plastic
requires 2 hours in plant A and 3 hours in plant B. If plant A is available 10 hours per day and
plant B is available 15 hours per day, how many tons of each type of plastic can be made daily so
that the plants are fully used?
5.16 A manufacturer makes 2-minute, 6-minute, 9-minute film developers. Each ton of 2minute developer requires 6 minutes in plant A and 24 minutes in plant B. Each ton of 6-minute
developer requires 12 minutes in plant A and 12 minutes in plant B. Each ton of 9-minute
developer requires 12 minutes in plant A and 12 minutes in plant B. If plant A is available 10
hours per day and plant B is available 16 hours per day, how many tons of each type of developer
can be produced so that the plants are fully used?
5.17 A dietician is preparing a meal consisting of foods A, B, and C. Each ounce of food A
contains 2 units of protein, 3 units of fat, and 4 units of carbohydrate. Each ounce of food B
contains 3 units of protein, 2 units of fat, and 1 unit of carbohydrate. Each ounce of food C
contains 3 units of protein, 3 units of fat, and 2 units of carbohydrate. If the meal must provide
exactly 25 units of protein, 24 unit of fat, and 21 units of carbohydrate, how many ounces of each
type of food should be used?
 cos  sin  
5.18
Let A  

  sin  cos  
a) Determine a simple expression for A2
b) Determine a simple expression for A3
c) Conjecture the form of a simple expression for Ak , k a positive integer.
d) Prove or disprove your conjecture in part (c).
5.19 Find two 2x2 nonzero matrices that their products is a zero matrix.
5.20
Find three 2x2 nonzero matrices A, B, C such that AB = AC.
5.21
 x  2 y  3z  6

Solve these linear systems: a) 2 x  3 y  2 z  14
3x  y  z  2

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Chapter 2. Algebra
 x  y  z  5

b)  x  y  z  1
4 x  2 y  z  7

 2 0 1 
b)  4 1 0
 2 3 1 
5.22
 4 3
Evaluate the determinant of a) 

 2 1
5.23
Use the Cramer’s rule to solve this system:
x  4 y  2z  2

 x  6 y  z  1
2 x  5 y  z  4

5.24
By graphing tell how many solutions these following linear systems have?
2 x  y  5
 x  3 y  1
 x  3 y  1
a) 
b) 
c) 
x  3y  6
x  3y  1
3x  9 y  3
5.25
1
0
A
 2

3
Compute the determinant of
0
1 2 3 
0 1 1

0 0 1
2
1
0 1 2
1 3 4
B
 2 0 1

 1 1 0
0
5 
1

0
Puzzles: Black and White hats
Cannibals ambush a safari in the jungle and capture three men. The cannibals give the men a
single chance to escape uneaten.
The captives are lined up in order of height, and are tied to stakes. The man in the rear can see
the backs of his two friends, the man in the middle can see the back the man in front, and the man
in front cannot see anyone. The cannibals show the men five hats. Three of the hats are black and
two of the hats are white.
Blindfolds are then placed over each man's eyes and a hat is placed on each man's head. The
two hats left over are hidden. The blindfolds are then removed and it is said to the men that if one
of them can guess what color hat he is wearing they can all leave unharmed.
The man in the rear who can see both of his friends' hats but not his own says, "I don't know".
The middle man who can see the hat of the man in front, but not his own says, "I don't know".
The front man who cannot see ANYBODY'S hat says "I know!"
How did he know the color of his hat and what color was it?
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Chapter 2. Algebra
ASSIGNMENT:
A) Work in groups to solve the following problems
1/ Prove that every subspace of Rn has a basis with a finite number of elements.
2/ A linearly independent set of vectors in Rn contains at most n vector.
3/ Determine x and y so that
 2x  y x  2 y
A
 is a diagonal matrix.
 4 x  8 y 3x  y 
1 2
.
2 
4/ Let B  
3
Find all 2 x2 matrices A such that AB = BA.
5/ Prove that ( AB)n  An Bn for all positive integer n, if AB = BA.
6/ Prove or disprove: If AB is a diagonal matrix then at least one of A or B is a diagonal
matrix.
7/ Prove or disprove: The product AB is a diagonal matrix if B is a diagonal matrix.
8/ Prove that if A is invertible then AT is invertible and  AT    A1 
1
9/ Prove that  An    AT 
T
T
n
10/ Prove or disprove: rank ( A)  rank ( A2 ) for all square matrices A over R.
11/ A square matrix A is said to be idempotent if A2  A . Prove that if A is idempotent then
either
det(A) = 1 or det (A) = 0.
12/ Prove or disprove each of the following statements.
a) If 1 and 2 are eigenvalues of A, then c11  c22 is an eigenvalues of A for any c1 , c2  F .
b) If x1 , x2 are eigenvectors of A then c1 x1  c2 x2 is an eigenvector of A for any c1 , c2  F .
13/ Let A be a invertible matrices, prove that if  is an eigenvalue of A, then  1 is an
eigenvalue of A1 .
14/ Prove that A and AT have the same eigenvalues.
B/ Topics for references
Each group choose one of the following topics for group assignment:
Topic 1: Matrix (types of matrices; operation on matrices; invertible matrices; rank of matrix)
Topic 2: Vector spaces (definition; bases of subspaces; isomorphism of vector spaces).
Topic 3: System of linear equations (definition of linear equation system; how to solve the
linear system; Cramer equation system).
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Chapter 2. Algebra
Topic 4: Determinants (definition; relation between determinant and the invertible matrix;
How to find the determinant of a 3x3matrix; 4x4 matrix; applications of determinant).
Topic 5: Eigenvalues and Eigenvectors (definition; how to figure out the eigenvalue and
eigenvector of a given matrix; give examples).
Topic 6: Real quadratic forms (definitions and examples; bilinear forms; reduction of real
quadratic form and give examples).
Topic 7: Some applications of linear algebra (at least 3 applications).
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