A gallery of linear maps

Six matrices, six ways to move a picture. Watch the letter F turn, grow, stretch, lean, flip and flatten.

Read the columns

In A matrix moves the plane a matrix was the table of where two arrows land. I call those arrows $\mathbf{e}_1=(1,0)$ and $\mathbf{e}_2=(0,1)$. A bold $\mathbf{e}$ with a number under it always means one of these arrows on this site, never the number $e$. The first column of a matrix is where $\mathbf{e}_1$ lands, and the second column is where $\mathbf{e}_2$ lands.

That one rule lets you read every picture in the gallery from its matrix. Step through the clip and, before each map plays, guess the picture from the two columns.

Turn. A turn by an angle $\theta$ anticlockwise moves $\mathbf{e}_1$ round the unit circle to $(\cos\theta,\sin\theta)$. The arrow $\mathbf{e}_2$ starts a quarter turn further on, and it lands at $(-\sin\theta,\cos\theta)$, a quarter turn ahead of $\mathbf{e}_1$‘s new place. So

The minus sign sits at the top right because $\mathbf{e}_2$ swings to the left. For 30°, which is $\pi/6\approx0.5236$ rad, the columns are $(0.866,\,0.5)$ and $(-0.5,\,0.866)$.

Scale. $\begin{bmatrix}1.5&0\\0&1.5\end{bmatrix}$ makes both arrows 1.5 times as long. The F grows and keeps its shape.

Stretch. $\begin{bmatrix}2&0\\0&1\end{bmatrix}$ doubles $\mathbf{e}_1$ and leaves $\mathbf{e}_2$ alone. The F becomes twice as wide and just as tall.

Shear. $\mathbf{A}_\text{shear}=\begin{bmatrix}1&1\\0&1\end{bmatrix}$ keeps $\mathbf{e}_1$ and sends $\mathbf{e}_2$ to $(1,1)$. Each row of the grid slides sideways by its height, and the F leans over.

Reflect. $\mathbf{A}_\text{refl}=\begin{bmatrix}0&1\\1&0\end{bmatrix}$ swaps the two arrows. The line $y=x$ acts as a mirror, and the F comes out back to front.

Project. $\mathbf{A}_\text{proj}=\begin{bmatrix}0.5&0.5\\0.5&0.5\end{bmatrix}$ sends both arrows to the same place, $(0.5,\,0.5)$. Every point $(x,y)$ goes to $\big(\tfrac{x+y}{2},\tfrac{x+y}{2}\big)$, the point of the line $y=x$ straight across from it at a right angle. The whole F is flattened onto the line.

Reading also works backwards. Draw where the two arrows should land, and their coordinates, written as columns, are the matrix.

The two rules

Every map in the gallery keeps grid lines straight, parallel and evenly spaced, and keeps the origin in place. Written as rules, a map $T$ of the plane is linear when, for all vectors $\mathbf{x}$, $\mathbf{y}$ and every number $c$,

The first rule says adding first and mapping first give the same answer. The second says the same for scaling. I write a map with braces, $T\{\cdot\}$, as the signal processing course writes a system, and the two rules are the same there.

These two rules are why the columns are enough. Every vector is a combination of the unit arrows, $\mathbf{x}=x_1\mathbf{e}_1+x_2\mathbf{e}_2$. Apply both rules:

So once you know where $\mathbf{e}_1$ and $\mathbf{e}_2$ land, you know where everything lands. The matrix of $T$ is those two landing places side by side:

Here is a check on the shear. $\mathbf{A}_\text{shear}$ sends $(1,1)$ to $(2,1)$. Doubling the input, $(2,2)$ goes to $(4,2)$, which is twice $(2,1)$, as the second rule says.

Not linear: a shift

Here is a move that looks harmless: slide the whole plane one step to the right. Every point $\mathbf{x}$ goes to $\mathbf{x}+(1,0)$. Watch what the grid does.

The grid lines stay straight, so the shift looks linear. But watch the origin: it moves, and a linear map never moves the origin. Both rules fail. Take $T\{\mathbf{x}\}=\mathbf{x}+(1,0)$:

• $T\{\mathbf{0}\}=(1,0)$, not $\mathbf{0}$. With $c=0$ the second rule would need $T\{\mathbf{0}\}=0\cdot T\{\mathbf{0}\}=\mathbf{0}$.
• $T\{\mathbf{x}+\mathbf{y}\}=\mathbf{x}+\mathbf{y}+(1,0)$ gets one shift, but $T\{\mathbf{x}\}+T\{\mathbf{y}\}=\mathbf{x}+\mathbf{y}+(2,0)$ gets two.

No 2×2 matrix can do a shift, because every matrix sends $\mathbf{0}$ to $\mathbf{0}$. Matrices in computer graphics shows the way round this.

Maps of space

In space a matrix has three columns: where $\mathbf{e}_1=(1,0,0)$, $\mathbf{e}_2=(0,1,0)$ and $\mathbf{e}_3=(0,0,1)$ land. Take a quarter turn about the vertical axis, the third one. It turns $\mathbf{e}_1$ to $(0,1,0)$ and $\mathbf{e}_2$ to $(-1,0,0)$, and it leaves $\mathbf{e}_3$, the axis itself, where it is:

It sends the point $(1,2,3)$ to

The first two numbers turn exactly as in the plane, and the height stays 3.

Worked example

1. From a picture to a matrix. A map sends $\mathbf{e}_1$ to $(0,2)$ and $\mathbf{e}_2$ to $(-1,0)$. Its columns are those two landing places:

It sends $(3,1)$ to $3\,(0,2)+1\,(-1,0)=(-1,6)$. By rows: $0\cdot3+(-1)\cdot1=-1$ and $2\cdot3+0\cdot1=6$. In words, this map turns everything a quarter turn anticlockwise and then doubles every height: the turn takes $\mathbf{e}_1$ to $(0,1)$, which doubles to $(0,2)$, and $\mathbf{e}_2$ to $(-1,0)$, whose height is 0.

2. The shift fails. $T\{\mathbf{x}\}=\mathbf{x}+(1,0)$ gives $T\{\mathbf{0}\}=(1,0)\ne\mathbf{0}$, so it is not linear.

3. The shear passes the scaling rule. $\mathbf{A}_\text{shear}(2,2)=(4,2)=2\,(2,1)=2\,\mathbf{A}_\text{shear}(1,1)$.

4. A reflection you write yourself. The mirror in the horizontal axis keeps $\mathbf{e}_1$ and flips $\mathbf{e}_2$ to $(0,-1)$, so its matrix is $\begin{bmatrix}1&0\\0&-1\end{bmatrix}$.

5. A turn by 30°. It sends $(1,0)$ to its first column, $(0.866,\,0.5)$.

Predict before you watch the gallery again: which of the six maps can’t be undone? The projection. It sends a whole line of points to each single point of $y=x$, so nothing can tell where a point started.

Where you’ll meet this

Image editors. Rotate, resize, skew and flip are these matrices, applied to the position of every pixel. Skew is the shear, and flip is a reflection.

Turning a phone. A phone’s motion sensor reports gravity in the sensor’s own axes. Turn the phone a quarter turn and the reading $(0.3,\,9.7)$ m/s² becomes $(-9.7,\,0.3)$ m/s² in the screen’s axes: $\mathbf{A}_\text{rot}(\theta)$ with $\theta=\pi/2$, a quarter turn, applied to the reading.

Complex numbers. Multiplying a complex number $a+jb$ by $\cos\theta+j\sin\theta$ gives $(a\cos\theta-b\sin\theta)+j(a\sin\theta+b\cos\theta)$. That is $\mathbf{A}_\text{rot}(\theta)$ applied to the point $(a,b)$, the rotate-and-scale picture of Complex numbers for signals in the signal processing course.

Reference card