1.1 Matrices and Systems of Linear Equations

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A linear equation can be defined as any equation that when graphed forms a straight line. Easy enough. An example of a linear equation is:

2x + y = 4

Any pair of values that satisfies this equation will for a straight line. When we put this equation with another:

-x + y = 1

This is a system of equations with two variables. We can solve this system to find the point or points where they intersect. This is called the solution. A system of equations can have either a unique solution, no solution, or infinite solutions.

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Unique Solution

This simply means that there is only one number for each of the variables that satisfies both equations. A graph of such a system would have one intersection point. Take our earlier equations:

2x + y = 4
-x + y = 1

This graph interstects at the point ( 1 , 2 ). This means that the x value of 1 and the y value of 2 satisfies both equations.

No Solution

Sometimes two equations never meet, which means they have no solution for the system. Take for example these equations:

-x + y =1
-2x + 2y = 4

These two lines, when graphed, are parallel and never meet or cross. There are no set of numbers that satisfy both equations.

Infinite Solution

When two equations have more than one solution, they often have infinite solutions.

-x + y=1
-3x + 3y = 3

These two equations make the same graph, meeting at every point. Which means that any solution that satisfies one will automatically satisfy the other.

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The equations we will be looking at, however, will not be simply two variable equations. We will be taking a look at multiple variable linear equations. Definition of linear equation - any equation that with n variables x₁, x₂, x₃,..., x_n that takes the form:

a₁x₁ + a₂x₂ + a₃x₃ + ... + a_nx_n = b

Where a₁, a₂, ... , a_n and b are constants. (Constants are numbers.)

And example of three linear equations with multiple variables is:

2x + 3y + 4z = 1
x - y + 2z = 3
4x + y + 8z = 0

(This system has no solution.)

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A matrix is a rectangular array of numbers. We use them to describe systems of linear equations. They are usually denoted by capital letters.

    [2 1 4]
A=[3 1 6]

This is a 2 x 3 matrix, meaning it has 2 rows and 3 columns. The rows are:

[2 1 4] and [3 1 6]

The columns are

[ 2 ]   [ 1 ]   [ 4 ]
[ 3 ]   [ 1 ]   [ 6 ]

A submatrix is obtained by eliminating either a row or column of the existing matrix. We denote them with a different letter. Submatrices of A are as follows:

P= [2 1 4]

R=[2 1]
    [3 6]

There can be many different submatrices for one matrix.

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If there are any questions, please post them in the comment area.

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