Download Group Assignment 2.

Group Assignment 2.
GROUP 1
I. Writing (1.5 ms.): Write about the following topic, give examples to illustrate and
finally give a brief translation in Vietnamese. (Notice: The translation is not more than 500
words).
Topic: Matrix (types of matrices; operation on matrices; invertible matrices; rank of
matrix)
II. Exercise (5 ms.):
1) Determine x and y so that
 2x  y x  2 y 
A
 is a diagonal matrix.
 4 x  8 y 3x  y 
2) 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 .
1 1 0 
3) Find all values of a for which the inverse of A  1 0 0 
1 2 a 
4) Prove or disprove the following statement: “For 3x3 matrix A, if B is the matrix
obtained by adding 5 to each entry of A, then det (B) = 5 + det (A).”
5) 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?
III. Reading: (1.5 ms.)
Read the following text give the brief translation in Vietnamese. Then choose
three properties and give their proofs.
A group is finite, or has finite order, if it contains a finite number of elements;
otherwise, the group is said to be infinite or to have infinite order. The order of a finite
group is the number of elements that it contains. If G is a group containing n elements, we
write |G| = n. The group 5 is a finite group of order 5; the integers
form an infinite
group under addition, and we sometimes write | |  .
A group G with the property that a b  b a for all a, b  G is called abelian or
commutative. Groups not satisfying this property are said to be nonabelian or
noncommutative.
Basic properties of group:
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Group Assignment 2.
1) The identity element in a group G is unique; that is, there exists only one element
e  G such that eg = ge = g for all g  G .
2) If g is any element in a group G, then the inverse of g, g 1 is unique.
3) Let G be a group. If a; b 2 G, then (ab)1  b1a 1
4) Let G be a group. For any a 2 G,  a 1   a
1
5) Let G be a group and a and b be any two elements in G. Then the equations ax = b
and xa = b have unique solutions in G.
6) If G is a group and a; b; c 2 G, then ba = ca implies b = c and ab = ac implies b = c.
IV. Grammar exercise (2 ms.)
Put the words in the following sentences in the correct order to make a
meaningful sentence, then translate each of the sentences into Vietnamese:
a) Variable/ unknown / is / a / letter / an /used / to / represent / quantity.
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.
GROUP 2
I. Writing (1.5 ms):
Write about the following topic give examples to illustrate and finally give a brief
translation in Vietnamese (Notice: The translation is not more than 500 words).
Topic: System of linear equations (Definition of linear equation system; How to solve
the linear system; Cramer equation system).
II. Exercise (5 ms.)
1 2
.
2 
1) Let B  
3
Find all 2 x2 matrices A such that AB = BA.
2) Prove or disprove the following statements: “ ( AB)n  An Bn for all positive integer n,
if AB = BA”.
3) Solve the matrix operation for x and y:
 2 3  y 7    10 20
   4 10    10 30 
x
5

 
 


a) 2  
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Group Assignment 2.
  3 x   y 2  0 2
 


5 0  1 3  6 3


b)
4) a) Find two 2x2 nonzero matrices that their products is a zero matrix.
b) Find two 3x3 nonzero matrices that their products is a zero matrix.
5) 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?
III. Reading: (1.5 ms.)
Read the following text give the brief translation in Vietnamese. Then give
examples of cyclic groups (which are finite or infinite).
Let G be a group and a be any element in G. Then the set
a  a k | k 
 is
a
subgroup of G. Furthermore, a is the smallest subgroup of G that contains a. For a  G ,
we call a the cyclic subgroup generated by a. If G contains some element a such that G
= a , then G is a cyclic group. In this case a is a generator of G. If a is an element of a
group G, we define the order of a to be the smallest positive integer n such that an = e, and
we write |a| = n. If there is no such integer n, we say that the order of a is infinite and write
| a |  to denote the order of a.
IV. Grammar exercises (2 ms.)
Choose the correct words to fill in the blanks.
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 .
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.
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Group Assignment 2.
GROUP 3
I. Writing: (1.5 ms) Write about the following topic give examples to illustrate and
finally give a brief translation in Vietnamese (Notice: The translation is not more than 500
words).
Topic: “Some applications of linear algebra (at least 3 applications).”
II. Exercise: (5 ms)
2
1) Let A  
 4
3
and f ( x)  x3  4 x2  5x  1 . Evaluate f ( A) .
1 
2) 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.
3) 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?
4) A girl, a boy, and a dog start walking down a road. They start at the same time, from
the same point, in the same direction. The boy walks at 5 km/h, the girl at 6 km/h.
The dog runs from boy to girl and back again with a constant speed of 10 km/h. The dog
does not slow down on the turn. How far does the dog travel in 1 hour?
5) Prove or disprove the following statement:
“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.”
III. Reading: (1.5 ms)
Read the following text and give the brief translation in Vietnamese. Then give
some example of group and its coset.
Let G be a group and H a subgroup of G. Define a left coset of H with representative
g  G to be the set gH  {gh | h  H } . Right cosets can be defined similarly by
Hg  {hg : h  H } . If left and right cosets coincide or if it is clear from the context to which
type of coset that we are referring, we will use the word coset without specifying left or
right.
IV. Grammar exercises (2ms.):
Choose the correct word to fill in the blanks. Then give a brief translation in
Vietnamese.
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
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Group Assignment 2.
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
2
above algebraic expression a is the coefficient of the  x  y  term.
GROUP 4
I. Writing: (1.5 ms.) Write about the following topic give examples to illustrate and
finally give a brief translation in Vietnamese (Notice: The translation is not more than 500
words).
Topic: “Real quadratic forms (definitions and examples; bilinear forms; reduction of
real quadratic form and give examples).”
II. Exercise: (5 ms.)
1) Prove or disprove the following statement: “If AB is a diagonal matrix then at least
one of A or B is a diagonal matrix.”
2) Prove or disprove the following statement:  An    AT 
T
n
3) Let A be a invertible matrices, prove that if  is an eigenvalue of A, then  1 is an
eigenvalue of A1 . Give example to illustrate this theorem.
 cos  sin  
4) Let A  

  sin  cos  
a) Determine a simple expression for A2 ; A3
b) Conjecture the form of a simple expression for Ak , k a positive integer
5) a) 5 years ago Kate was 5 times as old as her Son. 5 years hence her age will be 8
less than three times the corresponding age of her Son. Find their ages.
b) Using the numerals 1,7,7,7 and 7 (a "1" and four "7"s) create the number 100. As
well as the five numerals you can use the usual mathematical operations +, -, x, ÷ and
brackets (). For example: (7+1) × (7+7) = 112 would be a good attempt, but not right,
because it is not 100.
III. Reading (1.5 ms)
Read the following text give the brief translation in Vietnamese. Then give
examples of one infinite group and its subgroups and two finite groups and their
subgroups.
Subgroups:
Sometimes we wish to investigate smaller groups sitting inside a larger group.
The set of even integers 2  {...., 2, 0, 2, 4,...} is a group under the operation of
addition. This smaller group sits naturally inside of the group of integers under addition.
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Group Assignment 2.
We define a subgroup H of a group G to be a subset H of G such that when the group
operation of G is restricted to H, H is a group in its own right. Observe that every group G
with at least two elements will always have at least two subgroups, the subgroup consisting
of the identity element alone and the entire group itself. The subgroup H = {e} of a group
G is called the trivial subgroup. A subgroup that is a proper subset of G is called a proper
subgroup. In many of the examples that we have investigated up to this point, there exist
other subgroups besides the trivial and improper subgroups.
IV. Grammar exercises (2ms.):
Put these sentences in the correct order. Then give the brief translation.
____(1)___In order to communicate effectively, we must agree on the precise meaning
of the terms which we use.
_______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
GROUP 5
I. Writing (1.5ms.): Write about the following topic give examples to illustrate and
finally give a brief translation in Vietnamese (Notice: The translation is not more than 500
words).
Topic: Determinants (definition; relation between determinant and the invertible
matrix; How to find the determinant of a 3x3matrix; 4x4 matrix; applications of
determinant).
II. Exercise (5 ms.):
1 2 
1) Let A  
 . Find all the matrices B such that AB = BA.
0 1 
1 1 0 
2) Let A  0 1 1  . Evaluate Ak , k 
0 0 1 
3) 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
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Group Assignment 2.
$120 on each super desk and $70 on each standard desk. If x is the number of standard
desks and y is the number of super desks manufactured in a week, find the production of
each to give a maximum profit of $38000 per week?
1 1 0 
4) Find all values of a for which the inverse of A  1 0 0 
1 2 a 
Give the inverse matrix of A.
5) There are 2 trees in a garden (tree "A" and "B") and on the both trees are some
birds. The birds of tree A say to the birds of tree B that if one of you comes to our tree,
then our population will be the double of yours. Then the birds of tree B tell to the birds of
tree A that if one of you comes here, then our population will be equal to that of yours.
Now answer: How many birds in each tree?
III. Write about piecewise function and inverse function (2ms.) (Note: Give a brief
translation in Vietnamese not more than 400 words).
IV. Puzzles: (1.5 ms.)
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?
GROUP 6
I. Writing: (1.5 ms.) Write about the following topic give examples to illustrate and
finally give a brief translation in Vietnamese (Notice: The translation is not more than 500
words).
Topic: Eigenvalues and Eigenvectors (definition; how to figure out the eigenvalue and
eigenvector of a given matrix; give examples).
II. Exercise: (5ms.)
1 1 1
 x 0 0


1) Let A  0 1 1 , B  1 y 0 .Evaluate AB and B 2 A
0 0 1
 2 1 z 
2) Prove or disprove the following statement: “If A is invertible then AT is invertible
1
T
and  AT    A1  ”.
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Group Assignment 2.
3) Prove that: “Matrix A and AT have the same eigenvalues.”
4) Prove or disprove the following statement: “Let A be a lower triangular matrix.
Prove that A is singular if and only if some diagonal entry of A is zero.”
a b 
5) Find the inverse matrix of A  
 where a, b, c, d are not equal to zero.
c d 
III. Reading: (1.5ms.)
Read the following text give the brief translation in Vietnamese. Then give some
examples for the general linear group and orthogonal group.
The set of all n x n invertible matrices forms a group called the general linear group.
We will denote this group by GLn ( ) .The general linear group has several important
subgroups. The multiplicative properties of the determinant imply that the set of matrices
with determinant one is a subgroup of the general linear group. Stated another way,
suppose that det(A) = 1 and det(B) = 1. Then det(AB) = det(A) det(B) = 1 and
det( A1 ) 
SL(
n
1
 1 . This subgroup is called the special linear group and is denoted by
det A
).
Another subgroup of GLn ( ) is the orthogonal group. A matrix A is orthogonal if
A  AT . The orthogonal group consists of the set of all orthogonal matrices. We write
O(n) for the nx n orthogonal group. We leave as an exercise the proof that O(n) is a
subgroup of GLn ( ) .
1
IV. Grammar exercises (2ms.):
Put the word in the correct order to make the meaningful sentences. Then
translate each of the following sentences into Vietnamese.
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
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.
_____________________________________________________________________
Notice: The deadline for the second group assignment is 28/10/2011. Each
group should give the handout and send email to teacher.
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