Download Systems of Linear Equations and Matrices

Chapter 1
Systems of Linear
Equations and Matrices
1.1
Introduction to Systems of Linear Equations
In this section, we give the main de…nitions of the concepts studied in this
chapter. In particular, we explain what a system of linear equations is and we
give geometric interpretations of such systems. Systematic solutions of these
systems will be studied in the next section and beyond.
1.1.1
A First Example
Consider the problem of …nding the point of intersection between the two lines
y = x+1 and y = x+5 analytically. We begin by giving a name to the quantity
we are trying to …nd. Call P the point of intersection, and (x; y) its coordinates.
Since P is on the …rst line, its coordinates must satisfy the equation of the …rst
line. Therefore, we have y = x + 1 which we rewrite as x + y = 1. P is also on
the second line, therefore its coordinates must also satisfy the equation of the
second line. Therefore, we also have y = x + 5 which we rewrite x + y = 5.
We see that the coordinates of P must satisfy x + y = 1 and x + y = 5. We
write this as
x+y =1
x+y =5
This is an example of a system of linear equations. Finding the coordinates of
P amounts to solving this system. We will learn how to do this very soon.
We now turn to linear equations and present some de…nitions and notation
used with them.
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CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES
1.1.2
Linear Equations in n variables
De…nitions
We are all familiar with linear equations in two variables, the equation of the
line in dimension two. The most commonly used form is the slope-intercept form
y = mx + b. However, in this class, we will use the standard form, ax + by = c.
Similarly, a linear equation in three variable is of the form ax + by + cz = d.
This is the equation of a plane in 3-D.
We can continue this way with more variables. However, some problems use
equations having hundreds of variables. We will run out of letters very quickly.
To solve this problem, we use one letter for the variables, and one letter for the
constants. To di¤erentiate between the various variables and constants, we add
an index to the letter representing them. This way, we can write the general
form of a linear equation in n variables, for any positive integer n.
De…nition 1 A linear equation in n variables x1 ; x2 ; :::; xn has the form
a1 x1 + a2 x2 + ::: + an xn = b
Using sigma notation it can also be written
n
X
ai xi = b
i=1
a1 ; a2 ; :::; an are called the coe¢ cients of the equation; they are real numbers. a1 is called the leading coe¢ cient.
x1 ; x2 ; :::; xn are the variables of the equation. x1 is called the leading
variable.
b is called the constant term; it is also a real number.
Linear equations have no products or roots of variables. No variable is involved in trigonometric, exponential or logarithmic functions. Variables appear
only to the …rst power.
Example 2 The equations below are linear
1. x + y
p
2. 2x
3. x1
z = 20
y = sin(2)
2x2 + 3x3
4x4 = 5
Example 3 The equations below are not linear. Try to decide why.
p
1. 2x + y + 1 z = 1
2. ex + y = 1
3. sin x1 + cos x2 + x3 = 5
4.
1
1
+
=1
x1
x2
1.1. INTRODUCTION TO SYSTEMS OF LINEAR EQUATIONS
3
Solutions of a Linear Equation
De…nition 4 A solution of a linear equation in n variables is a sequence of
real numbers s1 ; s2 ; :::; sn arranged so that the equation is satis…ed when xi = si
for i = 1; 2; :::; n.
Example 5 The linear equation x1 + 2x2 = 4 is satis…ed when x1 = 0 and
x2 = 2. There are other solutions. In fact, this equation has in…nitely many
solutions.
De…nition 6 The set of all the solutions of a linear equation is called the solution set of that equation.
To …nd the solution set of a linear equation, we solve for one of the variables.
Let us look at some examples.
Example 7 Find the solution set of x1 + 2x2 = 4
We solve for x1 to obtain x1 = 4 2x2 . All the solutions can be obtained by
assigning a value to x2 and …nding the corresponding value for x1 .
De…nition 8 In the above example, x2 is called the free variable because it can
be anything. x1 is not free, its value depends on what x2 is.
Example 9 Find the solution set of x1 2x2 + x3 = 5
If we solve for x1 , we obtain x1 = 2x2 x3 + 5. Here, x2 and x3 are the free
variables. x1 is not free, its value is …xed, once x2 and x3 are known.
Parametric Representation of the Solution Set
One convenient way of representing the solution set of a linear equation is called
its parametric representation. It is obtained as follows. We assign a letter,
called a parameter, to each of the free variable. We express the variable which
is not free in terms of the parameters. The examples below illustrate this technique.
Example 10 Find the parametric representation of x1 + 2x2 = 4.
Solving for x1 gives x1 = 4 2x2 . The free variable is x2 . If we let x2 = t, then
we have x1 = 4 2t. We write the solution set as
x1 = 4 2t
x2 = t
where t can be any real number. This emphasizes the fact that x2 is free, and
x1 depends on what x2 is.
Example 11 Find the parametric representation of x1 2x2 + x3 = 5.
Solving for x1 gives x1 = 2x2 x3 + 5. Thus, x2 and x3 are the free variables.
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CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES
If we let x2 = t and x3 = s then we see that the parametric representation of
the solution set is
8
< x1 = 2t s + 5
x2 = t
:
x3 = s
where t and s can be any real numbers.
Remark 12 In the examples above, we solved for x1 every time. It does not
have to be that way. We did so because solving for x1 was the most convenient.
One can solve for any of the variables. Of course, the solution set will look
di¤ erent. However, it will still represent the same solutions.
We now turn to systems of linear equations.
1.1.3
Systems of Linear Equations
De…nition 13 A system of m linear equations in n variables is a set of
m equations, each equation being linear in n variables. It has the form:
8
a11 x1 + a12 x2 + ::: + a1n xn = b1
>
>
<
a21 x1 + a22 x2 + ::: + a2n xn = b2
(1.1)
:::
>
>
:
am1 x1 + am2 x2 + ::: + amn xn = bm
We used double subscripts on the coe¢ cients to indicate which variable as
well as which equation they correspond to. aij is the coe¢ cient of xj in the ith
equation.
A solution of a system of linear equations is a sequence of numbers s1 ; s2 ; :::; sn
that is a solution of each of the linear equations in the system.
Example 14 A solution of
x1 + x2 = 2
is (1; 1).
x1 x2 = 0
Recalling the …rst example we saw at the beginning of this handout, we see
that a geometric interpretation of a linear system of two equations in two variables gives us the intersection of the two lines described by the linear equations
of the system. We know that two lines can intersect in either 0 points (if they
are parallel), 1 point, or in…nitely many points (if the two linear equations of
the system correspond to the same line). This suggests that a system of two
linear equations in two variables can have 0; 1 or in…nitely many solutions. It
turns out that this is also true for larger systems. We state this as a theorem,
which will be proven later this semester. Before we state the theorem, we de…ne
a few terms.
De…nition 15 A system of linear equations is called consistent if it has at
least one solution. If it has no solution, it is called inconsistent.
1.1. INTRODUCTION TO SYSTEMS OF LINEAR EQUATIONS
5
Theorem 16 For a system of linear equations in n variables, precisely one of
the following happens:
1. The system has exactly one solution (consistent)
2. The system has in…nitely many solutions (consistent)
3. The system has no solution (inconsistent)
1.1.4
Augmented Matrices
The solution(s) of a system of m equations in n unknowns depends only on
the coe¢ cients which appear in the equations. It is often convenient to write
the system in a more compact form. We do so by putting the coe¢ cients in a
rectangular array in the same order they appear in the equations. if a variable
is missing in one of the equations, it means its coe¢ cient was 0. This is called
the augmented matrix for the system. Note the word matrix is the singular
of matrices. The augmented matrix for the system in equation 1.1 is
2
3
a11 a12
a1n
b1
6 a21 a22
a2n
b2 7
6
7
(1.2)
6 ..
..
..
.. 7
4 .
.
.
. 5
am1 am2
amn
bm
To write the augmented matrix, …rst make sure the following conditions are
satis…ed:
1. The variables must appear in the same order in each equation. If this is
not the case, rewrite the equation.
2. The constant term in each equation is to the right of the equal sign, all
the other terms are on the left.
Remark 17 You will notice that we left a space between the coe¢ cients and
the constant terms.
Example 18 Find the augmented matrix of the
8
< x1 + x2 + x3
2x1
2x2 + x3
:
x1 + x2 + 2x3
system
=
=
=
4
4
6
Answer: Since each equation satis…es the conditions given above, we simply
copy the coe¢ cients in a rectangular array. The augmented matrix is:
2
3
1 1 1
4
4 2
2 1
4 5
1 1 2
6
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CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES
Example 19 Find the augmented matrix of the system
8
< x y + 3z 4 = 0
2y + x = z 1
:
x+y z =8
Answer: First, we must write the equations in the correct format. The system
becomes
8
< x y + 3z = 4
x + 2y z = 1
:
x+y z =8
Now, we can write the augmented matrix:
2
1
1 3
4 1 2
1
1 1
1
1.1.5
3
4
1 5
8
Concept Review
Be able to recognize a linear equation in n variables.
Be able to …nd the solution set of a linear equation, and write it in parametric form.
Be able to recognize a system of linear equations.
Know the geometric meaning of a system of 2 linear equations.
Know what consistent and inconsistent linear systems are.
Be able to write the augmented matrix for a system.
1.1.6
Problems
1. On pages 6, 7 do # 1, 2, 3, 4, 5, 6, 7, 11, 12.