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 + ... + an*xn + b = 0- We say that x1,x2, … ,xn are the variables of the linear equation.
- We say that a1,a2, … ,an 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?
Gram Schmidt Process Visualization
