What is Linear Algebra? - Bigsigma Math Tutorials

Linear algebra is all about linear equations. In other words, In linear algebra, we investigate linear equations. How can we solve a group of linear equations? How can we represent a group of linear equations in matrices? When a group of linear equations has one solution, multiple solutions, or does not have a solution?

What is a linear equation?

First, what is linear equation?
any equation that can be put in the form:

a₁*x₁ + a₂*x₂ + a₃*x₃ + ... + a_n*x_n + b = 0

We say that x₁,x₂, … ,x_n are the variables of the linear equation.
We say that a₁,a₂, … ,a_n are the coefficients of the of the linear equation..

For example: 3x+2y = 10 is linear equation since it can put in the form 3x+2y + (-10) = 0

(3,2,-10) are the coefficients
x,y are the variables

We say that a sequence of numbers is a solution of the linear equation if replacing the variables with this group makes the equality true.

For example: Let’s see this definition 3x+2y = 10

(2 ,2) is a solution of the linear equation, since 3*2 + 2*2 = 10 is true.
(3, 0.5) is also a solution of the linear equation, since 3*3 + 2*0.5 = 10 is true.
However, (1,1) is not a solution of the linear equation since 3*1 + 2*1 = 10 is false.

As we can see in the above example, a single linear equation can have many solutions.

What is a group of linear equations?

Sometimes, we want to want to treat several linear equations as one entity : a group of linear equations.

We can extend the solution definition for a group of linear equations, We say that a sequence of numbers is a solution of a group of linear equation if when replacing the variables with this group make the equalizes true.

For the following group of linear equations 3x+2y = 10, x+y=4 :

(2,2) is solution since 3*2+2*2=10 is true and 2+2=4 is true
(3,0.5) is not a solution since 3+0.5=4 is false. It is not a solution, even thought 3*3+2*0.5 = 10 is true. In other words, It is not a solution since not all the equations in the group need to be true for the sequence.

What we investigate in linear algebra?

In linear algebra, we investigate linear equations.

How can we concisely represent linear equations and group of linear equations?
How can we find all the solutions of the group of linear equations?
What are algorithms can we use to solve the group of linear equations?
When a group of linear equations has one solution, many solutions, and when it does not have any solution?