Matrices, linear systems, and linear transformations

Connections to linear systems

We can now view a linear system as a single equation involving a matrix-vector product:

$$\left\{ \begin{aligned} a x + b y &= \alpha \\ c x + d y &= \beta \\ \end{aligned} \right. \quad\Leftrightarrow\quad \begin{bmatrix} a & b \\ c & d \end{bmatrix} \, \begin{bmatrix} x \\ y \end{bmatrix} \;=\; \begin{bmatrix} \alpha \\ \beta \end{bmatrix}.$$

In general, a system of $m$ equations involving $n$ unknowns can be written as a compact equation

$$A \mathbf{x} = \mathbf{b}$$

where $A$ is an $m \times n$ matrix that collects all the coefficients, $\mathbf{x}$ is a column vector containing $n$ entries $x_1,\dots,x_n$, each representing an unknown, and $\mathbf{b}$ is a column vector containing $m$ entries that are the right-hand-sides of the equations.

Matrix-vector products as transformations

For a given $2 \times 2$ matrix $A$, the function $\mathbf{v} \mapsto A \mathbf{v}$ can be considered as a transformation of vectors in $\mathbb{R}^2$. That is, the function $f : \mathbb{R}^2 \to \mathbb{R}^2$ given by $f(\mathbf{v}) = A \mathbf{v}$ is defined everywhere in $\mathbb{R}^2$ and sends vectors in $\mathbb{R}^2$ to vectors in $\mathbb{R}^2$.

Exercise. Consider matrices

$$\begin{aligned} A &= \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} & B &= \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix} & C &= \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \end{aligned}$$

What are the geometric interpretations of the transformation $\mathbf{v} \mapsto A \mathbf{v}$, $\mathbf{v} \mapsto B \mathbf{v}$, and $\mathbf{v} \mapsto C \mathbf{v}$?

In general, given a $m \times n$ matrix $A$, the function $f(\mathbf{v}) = A \mathbf{v}$ is a function $f : \mathbb{R}^n \to \mathbb{R}^m$. That is, the domain if $f$ is $\mathbb{R}^n$ and its codomain is $\mathbb{R}^m$ (the range may not be the entire $\mathbb{R}^m$).

Linearity

For a $m \times n$ matrix $A$, a vector $\mathbf{v} \in \mathbb{R}^n$, and a real number $r$, it is easy to verify that

$$\begin{aligned} A (r \, \mathbf{v}) &= r \, A \, \mathbf{v} \end{aligned}$$

Similarly, for a $m \times n$ matrix $A$ and two vectors $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$, we can also verify that

$$\begin{aligned} A (\mathbf{u} + \mathbf{v}) &= A \, \mathbf{u} \, + \, A \, \mathbf{v} \end{aligned}$$

Therefore, for a given $m \times n$ matrix $A$, the function $\mathbf{v} \mapsto A \mathbf{v}$ preserves vector addition and scalar multiplication and is hence called a linear function.

It is also easy to verify that for two $m \times n$ matrices $A$ and $B$ and a vector $\mathbf{v} \in \mathbb{R}^n$,

$$(A+B) \mathbf{v} = A \mathbf{v} + B \mathbf{v}.$$