Linear Algebra: Classifying and Solving Systems of Linear Equations, Study notes of Linear Algebra

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LINEAR ALGEBRA

Introduction to

This section aims to:

a. Classify equation in n variables as to linear or non-linear.

b. Use back-substitution and Gaussian elimination to solve a system of linear

equations.

c. Appreciate the importance of solving systems of linear equations.

CONCEPTS

The study of liner algebra demands familiarity with algebra, analytic geometry

and trigonometry. Recall from analytic geometry that the equation of the line in

two-dimensional space has the form

a

1

x +

a

2

y = b where

a

1

, a

2

∧ b are constants. This is called linear equation in two

variables.

 A linear equation in two variables x and y has the form

a

1

x

a

2

y = b

 A linear equation in three variables x, y and z has the form

a

1

x

a

2

y + a

3

z = b

 A linear equation in n variables has the form

a

1

x

1

a

2

x

2

• a

3

x

3

• … + a

n

x

n

= b

Linear equation have no product or roots of variables and no variables involved

in trigonometric, exponential, or logarithmic functions. Variables appear only to

the first power.

Example:

x

2

• 2y = 7

2. xy + z = 2

e

x

-2y = 4

4. ½ x + y – 3z = √

x

2

• 2x+ 1 = 0

Items 1, 2, 3 and 5 are not linear equation since it has an exponent of two which

made the equation nonlinear. On the other hand, item no.4 shows a linear

equation since it appear only to the first power.

SOLUTIONS AND SOLUTION SET

A solution of a linear equation in n variables is a sequence of n real numbers

s

1

s

2

, s

3

, … s

n

arranged to satisfy the equation when you substitute the value.

The set of all solutions of a linear equation is called a solution set , and when

you have found this set, you have solved the equation.

Example:

Find the values of

x

1

and

x

2

to satisfy the equation.

x

1

• 2 x

2

SOLUTION

x

1

=2 and

x

2

= 1 satisfy the equation

x

1

• 2 x

2

. Some other solutions are

x

1

= -4 and

x

2

= 4 , x

1

= 0 ∧ x

2

2 , and x

1

=− 2 ∧ x

2

PARAMETRIC REPRESENTATION OF A SOLUTION SET

Parametric representation is used to describe the entire solution set of a linear

equation.

Example:

Solve the linear equation

x

1

• 2 x

2

x

1

2 x

2

let x

2

be t

x

1

• 2 t

If t = 2

x

1

x

1

If t = 3

x

1

x

1

SYSTEMS OF LINEAR EQUATION

A system of m linear equations in n variables is a set of m equations, each of

which is a linear in the same n variables.

EXERCISES:

A. Identify the following equations as to linear or non-linear equations.

1. 5x + 3y = 15

e

x

• 2y = -

3 x

2

y + 3 y

2

= 2z + 4

4. √ 2x + 7y = 4
5. 7 y − 4 x = 10

B. Solve the linear equation using the parametric representation.

3y - x - z = 6

C. Solve the ff. using the parametric representation and plot the points of the linear

equation.

1. 2x + 3y = 24
2. 2x – 4y = 4
3. x – 2y = 1

D. Using the back substitution in row- echelon form, solve the linear system.

x + 2y + 3z = 9

x + 3y + 4z = 11

• 6y – 10z = -