System of Linear Equations

Sentence

• one piece of information
• a statement give you information
• e.g. a dog is black

System of Sentences

a system is consist of pieces of information aka. sentences e.g.

• a dog is black
• a cat has the same color with a dog

Equation

It's also a statement give you information kinda like sentence.

Linear Equation

equation like $2a+3b=5$

• only allow numbers/scalar be attached to variables
  • counter examples / non-linear equations
    • $a^2+b^2=10$
    • $sin(a)+b^5=15$
    • $2^a-3^b=0$
    • $ab^2+\frac{b}{a}-\frac{3}{b}-log(c)=4^a$

Linear Equation Representation

• 2 variables => lines in coordinate plane

  • system of linear equations => arrangements of lines in the plane
    • it enable you to visualize their solutions and (non-)singularity

• 3 variables => planes in space

• more variables => high-dimensional space

• matrix

  • an array of numbers
  • coefficients in the systems of equations

Singularity

definition

if sentences in a system are redundant or contradict to each other, they make the system singular. Otherwise, the system is complete and non-singluar which is the best.

relation with solution

complete, non-singular system has unique solution

redundant, singular system has infinite solutions

contradictory, singular system has no solution $*$ they can be proved in geometry way

Trick to identify singularity in geometry way

set constant in equations of a system to $0$

it might change the crossed point status(solution) but it won't change (non-)singularity

idenfity matrix singularity

linear dependence and independency

• linear dependency
  • second equation carries same information as the first one
  • determinant is $0$
• linear independency
  • no row is a multiple of the other one
  • determinant is not $0$