## ABSTRACT

We begin with solving systems of linear algebraic equations. You will use matrix methods often to solve problems in your engineering courses and in the real world, although a software package may hide this fact from you. Numerical approximations of differential equations, optimization, data analysis, and applications to vibrations and circuits have methods and algorithms that have at their core systems of linear algebraic equations and matrix methods. For the system of linear algebraic equations

⎧ ⎨

⎩

x1 − x2 + x3 = 0 −2x1 + 2x2 − x3 = 0

3x2 + 2x3 = 4

⎫ ⎬

⎭ (1.1)

in unknowns x1, x2, x3, adding two times the first equation to the second equation gives an “equivalent system,”

⎧ ⎨

⎩

x1 − x2 + x3 = 0 x3 = 0

3x2 + 2x3 = 4

⎫ ⎬

⎭ .