A matrix is a function that moves the plane

What does a 2×2 matrix do to the plane?

It assigns each point of the plane an image point. The map is completely determined by the images of two arrows, the unit steps to the right and up.

Before you start · 3 quick checks on vectors and distance, about 2 min

Setup: vectors, matrices and linear maps

A vector \(x = (x_1, x_2)\) in \(\mathbb{R}^2\) is a point of the plane, or the arrow from the origin to that point. The two unit vectors are \(e_1 = (1, 0)\) and \(e_2 = (0, 1)\).

A 2×2 matrix \(A\) has two columns. Its product with \(x\) takes \(x_1\) times the first column plus \(x_2\) times the second:

\[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \qquad Ax = x_1\begin{pmatrix} a \\ c \end{pmatrix} + x_2\begin{pmatrix} b \\ d \end{pmatrix} = \begin{pmatrix} ax_1 + bx_2 \\ cx_1 + dx_2 \end{pmatrix}. \]

So \(A\) is a function: it sends each point \(x\) of the plane to the point \(Ax\). A map \(T\) from \(\mathbb{R}^2\) to \(\mathbb{R}^2\) is called linear if

\[ T(u + v) = Tu + Tv \quad\text{and}\quad T(cu) = c\,Tu \qquad\text{for all vectors } u, v \text{ and every real number } c. \]

Why a pair (x₁, x₂) stands for a column here

In a matrix product the vector is a column, \(\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}\). To save space this page writes it on one line as \((x_1, x_2)\); it is still the same column, and the product is the one in the box above.

The same pair names a point and an arrow: the arrow from the origin to the point \((x_1, x_2)\). Adding two vectors puts the arrows tip to tail, and multiplying by a number scales its length, and reverses its direction when the number is negative.

1 See

Match the target grid

Drag the two blue dots, the tips of \(Ae_1\) and \(Ae_2\) (where \(e_1\) and \(e_2\) land), or type their coordinates below the figure, until your grid and F lie on the dashed target.

How the targets, the dots and the match work

Each target is the picture of the ordinary grid after a hidden matrix has moved the plane. The shaded region is the image of the unit square, the square with corners \(0\), \(e_1\), \(e_1 + e_2\) and \(e_2\), and the F drawn in it shows which way round it went.

Your own map is set by the two dots. Once you choose where \(e_1\) and \(e_2\) land, the page sends every other point by the rule of the setup and draws your grid and F. You can drag the dots, or type their coordinates in the four boxes below the figure; in a box the arrow keys change the value by 0.1. The dots sit on a grid of 0.1, between −2.8 and 2.8 across and between −2.2 and 2.2 up and down.

A match means that every point of the unit square lands within 0.2 of where the target sends it. Checking the corners \(e_1\), \(e_2\) and \(e_1 + e_2\) is enough: the miss at a point of the square is never larger than the largest miss at a corner.

Key point To copy a target you only had to place two dots, and the matrix that does it has those two vectors as its columns. Where a matrix sends \(e_1\) and \(e_2\) decides where it sends every point.

2 Try

Images of points

Answer each question to see the reason. The explorer below opens once you have answered all three.

Goals: place the dots

The explorer opens when you have answered the three questions above.

Key point \(Ax = x_1Ae_1 + x_2Ae_2\). One point fixes only one combination of the two columns. Two points that do not lie on one line through the origin fix both columns, and so the whole matrix.

3 Prove

A linear map is fixed by two vectors

Theorem

Let \(T\) be a linear map from \(\mathbb{R}^2\) to \(\mathbb{R}^2\): \(T(u + v) = Tu + Tv\) and \(T(cu) = c\,Tu\) for all vectors \(u, v\) and every real number \(c\). Write \(a_1 = Te_1\) and \(a_2 = Te_2\), and let \(A\) be the 2×2 matrix whose first column is \(a_1\) and whose second column is \(a_2\). Then for every \(x = (x_1, x_2)\),

\[ Tx = x_1a_1 + x_2a_2 = Ax. \]

So two linear maps that agree at \(e_1\) and at \(e_2\) agree at every point, and every linear map of the plane is a matrix.

Scratch work, not part of the proof.

What linearity does to the grid, and why a translation is not linear

Let \(A\) be a 2×2 matrix.

The origin stays put: \(A0 = 0 \cdot Ae_1 + 0 \cdot Ae_2 = 0\).
Lines go to lines or to points: the points \(p + td\), for real \(t\) and a direction \(d \ne 0\), go to \(Ap + tAd\). When \(Ad \ne 0\) that is the line through \(Ap\) in the direction \(Ad\). When \(Ad = 0\) the whole line lands on the single point \(Ap\); for example \(\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\) sends each vertical line to one point of the \(x_1\)-axis.
Parallel lines: take the lines through \(p + kq\) in one direction \(d\), for \(k = 0, 1, 2, \ldots\). If \(Ad \ne 0\), their images are the lines through \(Ap + kAq\) in the direction \(Ad\): parallel, and evenly spaced, because the points \(Ap + kAq\) are equal steps apart. If \(Aq\) is a multiple of \(Ad\), the steps run along the lines and all the images are one and the same line.
When \(Ad = 0\): each of those lines lands on a single point \(Ap + kAq\). These points are equal steps apart along one line, or all the same point when \(Aq = 0\) as well.
A translation is not linear: the map \(x \mapsto x + b\) with \(b \ne 0\) sends 0 to \(b\). Every matrix sends 0 to 0, so no matrix gives this map, and by the theorem it is not linear.

Key point Write \(x = x_1e_1 + x_2e_2\), use additivity and then homogeneity, and \(Tx = x_1Te_1 + x_2Te_2\) follows. The two images \(Te_1\) and \(Te_2\) are the columns of the matrix of \(T\).

4 Apply

Problems

Each problem has hints and a full solution. Where an answer is a list of numbers, type exactly that many numbers in order, separated by commas. Enter exact decimals such as 0.4 or fractions such as 2/5; each entry may differ from the exact answer by at most \(10^{-9}\).