Intro to Linear Algebra

  • Linear Algebra — Study of Matrices and their operations

  • Matrix — A rectangular array of numbers or expressions arranged in rows and columns, which can represent linear equations or transformations.

  • The operations on matrices include addition, subtraction, and multiplication, as well as finding the determinant and inverse of a matrix.

Linear Systems in Two Unknowns

6x+3y=27−5x+2y=06x+3y=27\\-5x+2y=0

Back 1 million years ago, we would use substitution, elimination, or graphing to solve these equations.

  • Linear algebra is linear because we are concerned with linear systems → Systems in which the variable are raised to the 1

Substitution

Elimination

Graphing

6x+3y=276x=27−3yx=27−3y66x + 3y = 27 \\ 6x = 27 - 3y \\ x = \frac{27 - 3y}{6}

Set up the equations to “remove” a variable

2(6x+3y)=2(27)3(−5x+2y)=3(0)2(6x+3y)=2(27)\\3(-5x+2y)=3(0)

  ⟹  −5(27−3y6)+2y=0\implies-5\left(\frac{27-3y}{6}\right)+2y=0

  ⟹  y=5\implies y=5

  ⟹  12x+6y=54−15x+2y\implies 12x + 6y = 54 \\ -15x + 2y

6x+3(5)=27  ⟹  x=26x+3(5)=27 \\ \implies x = 2