Linear Algebra Introduction

Let’s start with what linear equations are exactly, a linear equation is any equation that can be written in the form of equation 1.1(The three dots just mean it’s continuing the pattern) where $\alpha_n$ are known constants and $x_n$ are unknowns.

\[\alpha_1x_1 + \alpha_2x_2 + ... + \alpha_nx_n = b\ \text{ (1.1)}\]

Now that you know how linear equations look like, have you ever came across a set of equations in the form of equation 1.2, where every $\alpha$ and $d$ are constants, and $x_1$, $x_2$ and $x_3$ are unknowns. If you have, you might also know how to solve these set’s of equations. We won’t go into that right now but if you ever see a set of equations in this form you can use linear algebra!

\[\begin{array}{rcl} \alpha_1x_1 + \alpha_2x_2 + \alpha_3x_3 &=& b_1 \\ \alpha_4x_1 + \alpha_5x_2 + \alpha_6x_3 &=& b_2 \\ \alpha_7x_1 + \alpha_8x_2 + \alpha_9x_3 &=& b_3 \\ \text{(1.2)} \end{array}\]

We can rewrite this set of equation with matrices (don’t worry if you don’t understand how it works yet because we will get into that later.)

\[A = \begin{pmatrix} \alpha_1 & \alpha_2 & \alpha_3 \\ \alpha_4 & \alpha_5 & \alpha_6 \\ \alpha_7 & \alpha_8 & \alpha_9 \end{pmatrix}, \quad x = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}, \quad b = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} \ \text{ (1.3)}\]

If we then write $Ax = b$ we will have the exact same of set of equations also called a linear system as we had in 1.2. Here A is a matrix and, x is a vector just like b. In the next section we will explore how multiplication with matrices and vectors works so you can also see why Ax=b is the same as the linear system we had before.