Transcript Chapter 1

Chapter 1
Section 1.1
Introduction to Matrices and Systems
of Linear Equations
Linear Combinations
A linear combination of the n
variables 𝑥1 , 𝑥2 , 𝑥3 , ⋯ , 𝑥𝑛 is an
expression of the form given to the
right where 𝑎1 , 𝑎2 , 𝑎3 , ⋯ , 𝑎𝑛 are
known constants (numbers)
𝑎1 𝑥1 + 𝑎2 𝑥2 + 𝑎3 𝑥3 + ⋯ + 𝑎𝑛 𝑥𝑛
This is called a linear combination of the variables
𝑥1 , 𝑥2 , 𝑥3 , ⋯ , 𝑥𝑛 .
2𝑥 + 3𝑦 Is a linear combination of x and y
2𝑥1 + 4𝑥4 + 5𝑥6 Is a linear combination of 𝑥1 , 𝑥4 , 𝑥6
Linear Equations
An equation with n different variables 𝑥1 , 𝑥2 , 𝑥3 , ⋯ , 𝑥𝑛 is called a linear equation if it is
possible it write the equation (maybe using some equivalent algebraic rearrangement) as
a linear combination where 𝑎1 , 𝑎2 , 𝑎3 , ⋯ , 𝑎𝑛 are called the coefficients of the equation
being set equal to a constant b.
𝑎1 𝑥1 + 𝑎2 𝑥2 + 𝑎3 𝑥3 + ⋯ + 𝑎𝑛 𝑥𝑛 = 𝑏
7𝑥 + 2𝑦 − 3𝑧 = 9
Linear Equation
4𝑥1 − 2𝑥4 = 3 2 − 𝑥2 + 5𝑥4
Or 4𝑥1 + 3𝑥2 − 17𝑥4 = 6
Linear Equation
𝑥2 𝑥5 + 𝑥1 = 6
3 cos 𝛼 − 2 sin 𝛽 = 8
Nonlinear Equations
If the equation (maybe after some algebra) is anything other than a number times a
variable being added or subtracted it is nonlinear. No powers, roots, variables in the
denominator, products of variables, trig functions etc.
Systems of linear equations and their solution
A 𝑚 × 𝑛 system of linear equations is a set of m linear equations with n variables.
The numbers 𝑎𝑖𝑗 represent the coefficient of the jth variable in the ith equation. Such
a system can be expressed in a form given to the below.
𝑎11 𝑥1
𝑎21 𝑥1
⋮
𝑎𝑚1 𝑥1
+
+
+
𝑎12 𝑥2
𝑎22 𝑥2
⋮
𝑎𝑚2 𝑥2
+
+
⋯
⋯
+
+
+
⋯
+
𝑎1𝑛 𝑥𝑛
𝑎2𝑛 𝑥𝑛
⋮
𝑎𝑚𝑛 𝑥𝑛
=
=
=
𝑏1
𝑏2
⋮
𝑏𝑚
A 𝑚 × 𝑛 system of linear equations.
A solution to a system of equations (sometimes called a simultaneous solution) with n
variables is a set of numbers 𝑠1 , 𝑠2 , 𝑠3 , ⋯ , 𝑠𝑛 such that all the numbers satisfy all of the
m equations in the system.
𝑎11 𝑠1
𝑎21 𝑠1
⋮
𝑎𝑚1 𝑠1
+
+
+
𝑎12 𝑠2
𝑎22 𝑠2
⋮
𝑎𝑚2 𝑠2
+
+
⋯
⋯
+
+
+
⋯
+
𝑎1𝑛 𝑠𝑛
𝑎2𝑛 𝑠𝑛
⋮
𝑎𝑚𝑛 𝑠𝑛
=
=
=
𝑏1
𝑏2
⋮
𝑏𝑚
A solution to a 𝑚 × 𝑛 system of linear equations.
Linear Combinations of functions
Let 𝑓1 𝑥 , 𝑓2 𝑥 , ⋯ , 𝑓𝑛 𝑥 be a set of
n functions (not necessarily linear)
and 𝑐1 , 𝑐2 , ⋯ , 𝑐𝑛 a set of constants we
can form a linear combination of
functions as shown to the right.
𝑐1 𝑓1 𝑥 + 𝑐2 𝑓2 𝑥 + ⋯ + 𝑐𝑛 𝑓𝑛 𝑥
A linear combination of functions
𝑓1 𝑥 , 𝑓2 𝑥 , ⋯ , 𝑓𝑛 𝑥
2𝑥 2 + 3 cos 𝑥 + 5 𝑥
A linear combination of the functions
𝑥 2 , cos 𝑥 , 𝑥
Substitution to a Linear System
If all of the equations in a nonlinear system are linear combinations of the same functions
a substitution can be done to transform the nonlinear system into a linear system.
System:
𝑥 + 4 sin 𝑦 = 5
3 𝑥 − 2 sin 𝑦 = 8
Nonlinear System
Substitute:
𝑥1 = 𝑥
𝑥2 = sin 𝑦
New System:
𝑥1 + 4𝑥2 = 5
3𝑥1 − 2𝑥2 = 8
Linear System
Solving by Graphing
As the name would imply the graphs of linear equations are lines. The idea is to
graph both lines on the same graph carefully. Look at the point where the two lines
cross (or try to estimate it as best as you can) the x and y coordinates are the
simultaneous solutions to the system of equations.
 3x  2 y  4
Look at the previous example: 
 4 x  y  13
15
14
13
12
 3x  2 y  4
2 y  3x  4
11
4 x  y  13
10
9
y  4 x  13
8
7
y  x2
3
2
6
The coordinates of the point
the lines cross are (2,5)
5
4
slope is 3/2
slope is -4
3
2
y-intercept is 2
y-intercept is 13
1
-1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
-1
The problem that you run into with graphing to find the solutions is that it can be
very imprecise. When the solutions involve fractions or more than 2 or 3
variables this is very imprecise and not practical. This is why we will look at
other algebraic methods that tell you the simultaneous solutions.
15
 x  3

y  2
More graphing examples:
5
4
3
2
Solution (-3,2)
x = -3
Vertical line
at -3
y=2
-5
-4
1
-3
-2
-1
1
2
3
4
5
1
2
3
4
5
-1
-2
Horizontal
line at 2
-3
-4
-5
5
 6 x  2 y  8
 3x  y  1
4
More graphing examples: 
 6 x  2 y  8
2 y  6x  8
y  3x  4
3x  y  1
 y  3x  1
slope is 3
slope is 3
y-intercept is -4
y-intercept is -1
y  3x  1
3
2
1
-5
-4
-3
-2
-1
-1
-2
-3
-4
-5
These lines are parallel which means they do not intersect. This means there is no
simultaneous solution to the system of equations. A system of equations that has
no simultaneous solution we call inconsistent.
Systems of Equations and Augmented Matrices
Systems of equations can be represented with matrices in a certain way.
1. Each row corresponds to an equation.
2. Each column to a variable and the last column to the constants.
We write the variables on one side of the equation and the constants on the other. In
the matrix separate the variables from the constants with a line (sometimes dashed).
The entries of the matrix are the coefficients of the variables. It is important that if a
variable does not show up in an equation that means the coefficient is 0 and that
entry in the matrix is 0. The entries on the other side of the line are the constants.
3x  4 y  7

9 x  y  8
system of
equations
Sometimes algebra
might be needed to
change the equations
to a matrix.
3  4 7 


9
1
8


x  3 y  5z  8

 xz 9
 4 y  3z  1

Augmented
Matrix
 y  3x  2

2( x  3)  5 y  10
 3x  y  2

2 x  6  5 y  10
system of
equations
 3x  y  2

2 x  5 y  16
1 3  5 8 


1
0

1
9


0 4 3 1
Augmented
Matrix
 3 1  2 


2

5

16


Matrix Representation and Notation
The augmented matrix is one matrix
associated with the system of
equations. There is another matrix
which we refer to as the coefficient
matrix.
𝐴 = 𝑎𝑖𝑗
𝑎11
𝑎12
= ⋮
𝑎𝑚1
𝑎12
𝑎22
⋮
𝑎𝑚2
⋯
⋯
⋯
𝑎11 𝑥1
𝑎21 𝑥1
⋮
𝑎𝑚1 𝑥1
𝑎1𝑛
𝑎2𝑛
⋮
𝑎𝑚𝑛
The coefficient matrix for the system
The matrix B is the coefficient matrix A
"augmented" with the column of
constants. This is sometimes written as:
𝐴𝑏
Where b is the matrix to the right.
+
+
+
𝑎12 𝑥2
𝑎22 𝑥2
⋮
𝑎𝑚2 𝑥2
+
+
⋯
⋯
+
+
+
⋯
+
𝑎1𝑛 𝑥𝑛
𝑎2𝑛 𝑥𝑛
⋮
𝑎𝑚𝑛 𝑥𝑛
=
=
=
A 𝑚 × 𝑛 system of linear equations.
𝑎11
𝑎12
𝐵= ⋮
𝑎𝑚1
𝑎12
𝑎22
⋮
𝑎𝑚2
⋯
⋯
⋯
𝑎1𝑛 𝑏1
𝑎2𝑛 𝑏2
⋮
⋮
𝑎𝑚𝑛 𝑏𝑚
The augmented matrix for the system
𝑏1
𝑏
𝑏= 2
⋮
𝑏𝑚
𝑏1
𝑏2
⋮
𝑏𝑚