Linear Algebra I: Systems of Linear Equations, Assignments of Linear Algebra

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Systems of Linear Equations

MATH 322, Linear Algebra I

J. Robert Buchanan

Department of Mathematics

Spring 2007

Introduction

Linear Algebra: study of vector spaces and linear operators.

Vector Space Linear Operator

n × n matrices multiplication by a fixed matrix

continuous functions on [ a , b ] integration

differentiable functions differentiation

Examples

Example

Which of the following equations are linear equations?

2 x +

y

x 1 − x 2

• x 3

x 1 = x 2

x 1

x 2

x 1

• 2 x 2
• 3 cos x 3

Solutions

Definition

A solution to a linear equation

a 1 x 1

• a 2 x 2
• · · · + a n x n = b

is a set of numbers s 1 , s 2 ,... , s n such that

a 1 s 1

• a 2 s 2
• · · · + a n s n = b.

Example

Example

Find a solution to the following system of linear equations.

3 x 1 − 2 x 2 = − 1

4 x 1

• 5 x 2

Note: there exists only one solution to the system above (we

say the solution is unique ). Not every linear system will have a

unique solution.

Graphical Interpretation

Consider the generic linear system:

a 11 x 1

• a 12 x 2 = b 1

a 21 x 1

• a 22 x 2 = b 2

The graph of each equation is a straight line.

The solution to the system (if any) will be the intersection

of the straight lines.

Remarks

A system of equations with no solution is said to be

inconsistent.

A system of equations with at least one solution is said to

be consistent.

These possibilities hold for systems of equations with any

number of unknowns.

Augmented Matrices

Every system of linear equations can be written in the form of

an augmented matrix.

a 11 x 1 + a 12 x 2 + · · · + a 1 nxn = b 1

a 21 x 1

• a 22 x 2
• · · · + a 2 n x n = b 2

a m 1 x 1

• a m 2 x 2
• · · · + amnxn = bm

becomes 

a 11 a 12 · · · a 1 n b 1

a 21 a 22 · · · a 2 n b 2

a m 1 a m 2 · · · a mn b m

Example

Example

Using elementary row operations on the appropriate

augmented matrix, find the solution to the following system of

equations.

2 x 1

• 2 x 3

3 x 1 − x 2

• 4 x 3

6 x 1

• x 2 − x 3

Homework

Read Section 1.1 and work exercises 1–13 odd.