Every linear map of the plane is decided by where it sends the two standard arrows, and a handful of kinds recur everywhere: turning, mirroring, flattening onto a line, stretching and sliding. Each is one small matrix, and its determinant says what it does to areas.

A slide projector can turn a picture, flip it, stretch it or throw its shadow onto a wall at an angle. Straight lines stay straight and the centre stays put, and those two facts are what make each of these a linear map.

A quarter turn anticlockwise sends $\mathbf{v}=(3,1)$ to $T\mathbf{v}=(-1,3)$: the length is kept and the two arrows meet at a right angle. It sends $\mathbf{e}_1$ to $(0,1)$ and $\mathbf{e}_2$ to $(-1,0)$, so its matrix is $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$.

Reflection in the line $y=x$ swaps the two coordinates: $\mathbf{v}=(3,1)$ goes to $T\mathbf{v}=(1,3)$, and the dashed segment joining them crosses the mirror at a right angle. Its matrix has determinant : the map turns the plane over.

A linear map $T$ of the plane is the map of the $2\times2$ matrix whose columns are $T\mathbf{e}_1$ and $T\mathbf{e}_2$. It sends the unit square to the parallelogram on those two columns, so it multiplies every area by , and it turns the plane over when . The kinds below are the building blocks: every invertible map of the plane is a product of them.

• Rotation by the angle $\theta$ anticlockwise: $R_\theta$ above, with determinant $\cos^2\theta+\sin^2\theta=1$.

Projection onto the $x$-axis keeps the first coordinate and drops the second: $\mathbf{v}=(2,3)$ goes to $T\mathbf{v}=(2,0)$. Every point of the dashed line lands on the same place, so the map cannot be undone, and its matrix has determinant .

A rotation about the origin is linear: turning a parallelogram turns its diagonal with it, and turning a stretched arrow gives the stretched turned arrow. So it has a matrix, whose columns are where $\mathbf{e}_1$ and $\mathbf{e}_2$ go. $\mathbf{e}_1$ lands on the point of the unit circle at angle , which is by the definition of the cosine and sine; is a quarter turn further on.

Turning a parallelogram turns its diagonal with it, and turning a stretched arrow gives the stretched turned arrow: the rotation is linear, so it has a matrix.

$\mathbf{e}_1$ is the point of the unit circle at angle $0$. Turned by $\theta$ it is the point at angle $\theta$, whose coordinates are the cosine and sine of $\theta$ by definition.

$\mathbf{e}_2$ is the point at angle $90^\circ$, so it lands at the angle $\theta+90^\circ$.

A quarter turn sends a point $(a,b)$ to $(-b,a)$, so the point at angle $\theta+90^\circ$ is .

The images of the two standard vectors are the columns of the matrix.

To find the matrix of a map of the plane, follow $\mathbf{e}_1=(1,0)$ and $\mathbf{e}_2=(0,1)$. Their images, written as columns, are the matrix.

Projection onto the $x$-axis, $\begin{pmatrix}1&0\\0&0\end{pmatrix}$, sends every point of a vertical line to one and the same point. Whole directions are crushed, so no map can undo it, and the unit square is flattened into a segment of area .

• The columns of the matrix of a map of the plane are $T\mathbf{e}_1$ and $T\mathbf{e}_2$.
• Rotation: $R_\theta$, determinant . Reflection: determinant . Projection onto a line: determinant .

Great work! You now know how to:

• write the matrix of a map of the plane from where it sends the standard vectors;
• recognise rotations, reflections, projections, scalings and shears;
• use the determinant as the area factor and the sign of orientation;
• compose maps of the plane by multiplying their matrices.