Matrices as Transformations of the Plane

A matrix acting on a vector

Write a vector as a column and multiply it by a $2 \times 2$ matrix. The result is another vector:

$$\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} ax + by \\ cx + dy \end{bmatrix}$$

Try it with a particular matrix:

$$\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 2 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \end{bmatrix}$$

The output has the same length as the input and points $90°$ further counterclockwise. This matrix does that to every vector. It is a rotation, and a matrix is a rule for moving the whole plane at once.

Why the columns tell the whole story

Feed the matrix the two simplest vectors, $\langle 1, 0\rangle$ and $\langle 0, 1\rangle$:

$$\begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} a \\ c \end{bmatrix} \qquad \begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} b \\ d \end{bmatrix}$$

The columns are where those two vectors land. And every vector is built from those two: $\langle x, y\rangle$ is $x$ copies of $\langle 1, 0\rangle$ plus $y$ copies of $\langle 0, 1\rangle$. A matrix respects that building, so the image is $x$ copies of the first column plus $y$ copies of the second.

Two columns therefore decide where every point of the plane goes. That also gives the fastest way to build a matrix: find where $\langle 1, 0\rangle$ and $\langle 0, 1\rangle$ go, and write them down as columns.

A catalog of transformations

Transformation	$\langle 1, 0\rangle$ goes to	$\langle 0, 1\rangle$ goes to	Matrix
rotate $90°$ counterclockwise	$\langle 0, 1\rangle$	$\langle -1, 0\rangle$	$\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$
reflect in the $x$-axis	$\langle 1, 0\rangle$	$\langle 0, -1\rangle$	$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$
reflect in $y = x$	$\langle 0, 1\rangle$	$\langle 1, 0\rangle$	$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
stretch by $2$ across, $3$ up	$\langle 2, 0\rangle$	$\langle 0, 3\rangle$	$\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$
shear sideways	$\langle 1, 0\rangle$	$\langle 1, 1\rangle$	$\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$

These are the transformations from geometry, now carried out by a single multiplication.

Area and the determinant

The unit square has corners $(0, 0)$, $(1, 0)$, $(1, 1)$ and $(0, 1)$. A matrix sends its two sides to the two columns, so the square becomes the parallelogram built on the columns.

Take $T = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}$. Its columns are $\langle 2, 0\rangle$ and $\langle 1, 3\rangle$.

The parallelogram has base $2$ and height $3$, so its area is $6$. The determinant is $2 \cdot 3 - 1 \cdot 0 = 6$ as well, and that is not a coincidence.

The absolute value of the determinant is the factor by which the transformation multiplies areas. The unit square has area $1$ and becomes a region of area $\lvert\det\rvert$. Every other shape, being built from tiny squares, scales by the same factor.

$\det$	Effect on the plane
$\lvert\det\rvert > 1$	areas grow
$0 < \lvert\det\rvert < 1$	areas shrink
negative	the plane is also flipped, like a mirror image
$0$	the plane collapses onto a line or a point

The last row explains the inverse rule. A matrix with determinant $0$ squashes the plane flat, sending many points to the same place, and nothing can pull them apart again. That is why it has no inverse.

Composing transformations

Doing one transformation and then another is a matrix product, with the first transformation on the right. Reflecting in the $x$-axis and then rotating $90°$ is $R F$:

$$R F = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$

That is a reflection in $y = x$. Doing them in the other order gives $FR = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$, a reflection in $y = -x$ — a different result, which is the non-commutative product seen from the geometric side.

Worked examples

Common mistakes

Practice problems