From systems of linear equations to matrix equations

Let’s start with a system of linear equations \(a_{11} x_1 + a_{12}x_2 + \cdots a_{1m} x_m = b_1 \\ a_{21} x_1 + a_{22}x_2 + \cdots a_{2m} x_m = b_2 \\ \vdots \\ a_{n1} x_1 + a_{n2}x_2 + \cdots a_{nm} x_m = b_n \\\)

If we set \(A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1m} \\ a_{21} & a_{22} & \cdots & a_{2m} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nm} \end{pmatrix} \ , \ \mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_m \end{pmatrix} \ , \ \mathbf{b} = \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{pmatrix}\) then the system of can be captured using the simple matrix equation \(A \mathbf{x} = \mathbf{b}.\)

Similarly, we can reverse the direction. Writing out $A \mathbf{x} = \mathbf{b}$ explicitly will yield the original system of linear equations. Thus, we lose no information when using the matrix language.

Augmented matrices

An extended notation for matrix equations will be useful, particularly for solving linear systems. Given a matrix equation $A \mathbf{x} = \mathbf{b}$, the augmented matrix is denoted \(\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1m} &\bigg| & b_1 \\ a_{21} & a_{22} & \cdots & a_{2m} &\bigg| & b_2 \\ \vdots & \vdots & \ddots & \vdots &\bigg| & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nm} &\bigg| & b_n \end{pmatrix}\)

We will write $(A \mid b)$ to denote the augmented matrix. We will often talk about the set of solutions to a linear system $(A \mid b)$ and we will use the notation $\mathcal Z(A \mid b)$ for this set. When $\mathbf{b} = \mathbf{0}$ we set \(\mathcal Z(A) = \mathcal Z(A \mid \mathbf{0}).\) The set $\mathcal Z(A)$ is often called the null space of $A$. (Hence you will see the notation $\mathcal N(A)$ elsewhere.)

Our next task is to understand how we can enhance our understanding of solutions of linear systems by using matrices. We will study which operations on (augmented) matrices can simplify the matrix into a canonical form without losing information about the solution set of the linear system.