3.2.2 Matrices as linear maps

A matrix moves the plane

Lesson 3.2.1 multiplied a matrix $A$ by a vector $\mathbf{x}$ . Now read that product as a function: $A$ takes each point $\mathbf{x}$ of the plane and sends it to $A\mathbf{x}$ . A function that takes vectors to vectors is called a map, and $A\mathbf{x}$ is the image of $\mathbf{x}$ .

Take the matrix

H = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}

It sends $(x_1, x_2)$ to $(x_1 + x_2,\ x_2)$ : each point slides sideways by its own height. This map is called a shear.

point $\mathbf{x}$	$(0, 0)$	$(1, 0)$	$(0, 1)$	$(1, 1)$	$(-1, 2)$	$(0, -1)$
image $H\mathbf{x}$	$(0, 0)$	$(1, 0)$	$(1, 1)$	$(2, 1)$	$(1, 2)$	$(-1, -1)$

Points on the horizontal axis, the $x_1$ -axis, stay where they are. Points above it move right, and points below it move left, each by its height. The unit square, with corners $(0, 0)$ , $(1, 0)$ , $(1, 1)$ and $(0, 1)$ , leans over into a parallelogram.

What makes a map linear

A map $T$ is linear when it keeps sums and scalings. Adding two vectors and then mapping gives the same as mapping each and then adding. Scaling by a number $c$ and then mapping gives the same as mapping and then scaling.

Rule: A linear map

\begin{gathered} T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \\[4pt] T(c\,\mathbf{u}) = c\,T(\mathbf{u}) \end{gathered}

Together these say that $T$ keeps linear combinations (lesson 3.1.1): $T(a\mathbf{u} + b\mathbf{v}) = a\,T(\mathbf{u}) + b\,T(\mathbf{v})$ . Every matrix gives a linear map: $A(\mathbf{u} + \mathbf{v}) = A\mathbf{u} + A\mathbf{v}$ and $A(c\mathbf{u}) = c\,A\mathbf{u}$ .

Check it on the shear with $\mathbf{u} = (1, 2)$ and $\mathbf{v} = (3, -1)$ . Adding first, $H(4, 1) = (5, 1)$ . Mapping first, $H\mathbf{u} + H\mathbf{v} = (3, 2) + (2, -1)$ , the same $(5, 1)$ .

Scaling works too: $H(3\mathbf{u}) = (9, 6)$ , which is $3 \cdot (3, 2)$ , three times $H\mathbf{u}$ .

Linearity shows in the picture. The origin never moves, since $T(\mathbf{0}) = T(0\,\mathbf{u}) = 0\,T(\mathbf{u}) = \mathbf{0}$ . Straight lines stay straight, and evenly spaced parallel grid lines stay parallel and evenly spaced (or collapse together).

Two maps that are not linear:

A shift, $T(\mathbf{x}) = \mathbf{x} + (1, 0)$ . Then $T(\mathbf{u} + \mathbf{v}) = (5, 1)$ but $T(\mathbf{u}) + T(\mathbf{v}) = (6, 1)$ : the shift is counted twice. It also moves the origin to $(1, 0)$ , so no matrix can shift the plane.
Squaring each component. $T(2\mathbf{u}) = (4, 16)$ , but $2\,T(\mathbf{u}) = (2, 8)$ .

(5, 1)

import numpy as np H = np.array([[1, 1], [0, 1]]) u = np.array([1, 2]) v = np.array([3, -1]) print(H @ (u + v), H @ u + H @ v) print(H @ (3 * u), 3 * (H @ u)) def shift(x): return x + np.array([1, 0]) def square(x): return x**2 print(shift(u + v), shift(u) + shift(v), shift(np.array([0, 0]))) print(square(2 * u), 2 * square(u))

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The columns are where e₁ and e₂ go

Write $\mathbf{e}_1 = (1, 0)$ and $\mathbf{e}_2 = (0, 1)$ , the standard basis of the plane. Every vector is a combination of them: $(x_1, x_2) = x_1\mathbf{e}_1 + x_2\mathbf{e}_2$ .

So for a linear map, $T(\mathbf{x}) = x_1\,T(\mathbf{e}_1) + x_2\,T(\mathbf{e}_2)$ . That is the column picture of lesson 3.2.1, with columns $T(\mathbf{e}_1)$ and $T(\mathbf{e}_2)$ . Once you know where two arrows go, you know where everything goes.

Rule: The columns are the images of the basis

\begin{gathered} A\mathbf{x} = x_1\,A\mathbf{e}_1 + x_2\,A\mathbf{e}_2 \\[4pt] \text{column } j \text{ of } A = A\mathbf{e}_j \end{gathered}

Every linear map of the plane is a $2 \times 2$ matrix. To find it, see where $\mathbf{e}_1$ and $\mathbf{e}_2$ land, and write those two images as the columns.

The matrix $A = \begin{pmatrix} 2 & 1 \\ -1 & 3 \end{pmatrix}$ of lesson 3.2.1 sends $\mathbf{e}_1$ to $(2, -1)$ and $\mathbf{e}_2$ to $(1, 3)$ , its two columns. So $A(3, 2) = 3\,(2, -1) + 2\,(1, 3) = (8, 3)$ .

Going the other way, suppose a map sends $\mathbf{e}_1$ to $(1, 2)$ and $\mathbf{e}_2$ to $(-1, 1)$ . Its matrix has those columns, $\begin{pmatrix} 1 & -1 \\ 2 & 1 \end{pmatrix}$ , and it sends $(2, 3)$ to $2\,(1, 2) + 3\,(-1, 1) = (-1, 7)$ . For the shear, $\mathbf{e}_1$ stays put and $\mathbf{e}_2$ moves to $(1, 1)$ .

The explorer starts on the shear. Its two coloured arrows are $A\mathbf{e}_1$ and $A\mathbf{e}_2$ , the columns, and the shaded shape is the image of the unit square. Slide the entry $b$ from 1 to 0.5, then to $-1$ , and watch the grid lean the other way.

Then try the rotation and the reflection, and drag the probe $\mathbf{x}$ to see $A\mathbf{x}$ . The readout also shows two numbers that lesson 3.4.1 and module 3.6 explain.

Explore

Pick a matrix or set its entries. Its first column is where e₁ lands and its second where e₂ lands; the whole grid follows. Then drag the probe x and look for directions that A only stretches.

a (row 1, column 1) = 1

b (row 1, column 2) = 1

c (row 2, column 1) = 0

d (row 2, column 2) = 1

Probe x = (1, 2) (drag its tip)Show eigenvector lines

Matrix: $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$
Columns: Ae₁ = (1, 0), Ae₂ = (1, 1)
det A: 1 × 1 − 1 × 0 = 1: every area is multiplied by 1.
Probe: Ax = (3, 2): a different direction.
Eigenvalues: one, λ = 1

A\mathbf{e}_1

import numpy as np H = np.array([[1, 1], [0, 1]]) for p in [(0, 0), (1, 0), (0, 1), (1, 1), (-1, 2), (0, -1)]: print(p, H @ np.array(p)) A = np.array([[2, 1], [-1, 3]]) e1, e2 = np.array([1, 0]), np.array([0, 1]) print(A @ e1, A @ e2) print(A @ np.array([3, 2]), 3 * (A @ e1) + 2 * (A @ e2)) M = np.column_stack([[1, 2], [-1, 1]]) print(M) print(M @ np.array([2, 3]))

⌘/Ctrl + Enter runs the cell

A gallery of maps

Five kinds of map appear again and again. Each one is fixed by where it sends $\mathbf{e}_1$ and $\mathbf{e}_2$ ; the examples use the point $(3, 2)$ .

The unit square, the grid and the images of $\mathbf{e}_1$ and $\mathbf{e}_2$ under six maps. The two arrows in each panel are the columns of its matrix.

Stretch. $\begin{pmatrix} 2 & 0 \\ 0 & 0.5 \end{pmatrix}$ doubles every horizontal distance and halves every vertical one: $(3, 2)$ goes to $(6, 1)$ . A negative entry also turns arrows round: $\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$ scales every vector by $-1$ (lesson 3.1.1).
Shear. $\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$ slides each point sideways by $k$ times its height. With $k = 1$ , $(3, 2)$ goes to $(5, 2)$ .
Rotation. A turn through the angle $\theta$ , anticlockwise about the origin, keeps every length. It sends $\mathbf{e}_1$ to $(\cos\theta, \sin\theta)$ , the point at angle $\theta$ on the unit circle (lesson 1.7.1). It sends $\mathbf{e}_2$ a quarter turn further on, to $(-\sin\theta, \cos\theta)$ ; a quarter turn takes $(a, b)$ to $(-b, a)$ , as in lesson 1.7.2.
Reflection. $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ flips the plane over the $x_1$ -axis: $(3, 2)$ goes to $(3, -2)$ . $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ flips it over the line $x_2 = x_1$ by swapping the components: $(3, 2)$ goes to $(2, 3)$ .
Projection onto an axis. $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ keeps the first component and drops the second. It sends $(3, 2)$ to $(3, 0)$ , and $(3, -5)$ there too. The whole plane lands on one line, and many points share an image; lesson 3.2.3 shows that such a map cannot be undone.

Rule: Rotation through θ

R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

The first column is the image of $\mathbf{e}_1$ and the second the image of $\mathbf{e}_2$ . For a quarter turn, $\theta = 90°$ , this is $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ .

The quarter turn sends $(3, 2)$ to $(-2, 3)$ , the same move as $i(3 + 2i) = -2 + 3i$ in lesson 1.7.2. A turn through 30° sends it to $(1.5981, 3.2321)$ . Both points lie 3.605551 from the origin, so the length is kept.

Maps change areas too. The projection squashes the unit square to a segment of area 0, while the shear and this stretch keep its area at 1. The factor by which a map multiplies every area is its determinant (lesson 3.4.1).

(3, 2)

import numpy as np def rotation(degrees): t = np.radians(degrees) return np.array([[np.cos(t), -np.sin(t)], [np.sin(t), np.cos(t)]]) x = np.array([3, 2]) maps = { "stretch": np.array([[2, 0], [0, 0.5]]), "shear": np.array([[1, 1], [0, 1]]), "rotate 90": np.array([[0, -1], [1, 0]]), "reflect in x1-axis": np.array([[1, 0], [0, -1]]), "reflect in x2 = x1": np.array([[0, 1], [1, 0]]), "project on x1-axis": np.array([[1, 0], [0, 0]]), } for name, A in maps.items(): print(name, A @ x) R = rotation(30) print(np.round(R, 6)) print(np.round(R @ x, 4), round(float(np.linalg.norm(x)), 6), round(float(np.linalg.norm(R @ x)), 6))

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One map after another

Apply $B$ first and then $A$ : $\mathbf{x}$ goes to $B\mathbf{x}$ , then to $A(B\mathbf{x})$ . Brackets can move in a product (lesson 3.2.1), so $A(B\mathbf{x}) = (AB)\mathbf{x}$ : the single matrix $AB$ does both steps.

Rule: One map after another

A(B\mathbf{x}) = (AB)\,\mathbf{x}

$AB$ means " $B$ first, then $A$ ". Read a product from right to left: the matrix next to $\mathbf{x}$ acts first, as in the composition $f \circ g$ of lesson 1.1.3.

Take the quarter turn $R = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ and the reflection $F = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ , and follow $(3, 2)$ both ways.

order	after the first map	after the second	one matrix
rotate, then reflect	$R(3, 2) = (-2, 3)$	$(-2, -3)$	$FR$
reflect, then rotate	$F(3, 2) = (3, -2)$	$(2, 3)$	$RF$

The products agree with the table:

\begin{gathered} FR = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \\[6pt] RF = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \end{gathered}

$RF$ is the reflection in the line $x_2 = x_1$ from the gallery. $FR$ is the reflection in the line $x_2 = -x_1$ . Both are reflections, but in different lines, so the order matters.

Some pairs give the same result in either order. Turning through 30° and then 60° is a turn through 90°, and so is 60° then 30°. Repeating a map multiplies it by itself:

Two quarter turns make a half turn, $RR = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$ : every vector is scaled by $-1$ , as $i^2 = -1$ in lesson 1.7.2.
Three shears make a shear by 3: $HHH = \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}$ .

A stretch and a shear do not commute. With $S = \begin{pmatrix} 2 & 0 \\ 0 & 0.5 \end{pmatrix}$ , shearing first gives $SH = \begin{pmatrix} 2 & 2 \\ 0 & 0.5 \end{pmatrix}$ , while stretching first gives $HS = \begin{pmatrix} 2 & 0.5 \\ 0 & 0.5 \end{pmatrix}$ .

(3, 2)

import numpy as np def rotation(degrees): t = np.radians(degrees) return np.array([[np.cos(t), -np.sin(t)], [np.sin(t), np.cos(t)]]) R = np.array([[0, -1], [1, 0]]) # rotate 90 degrees F = np.array([[1, 0], [0, -1]]) # reflect in the x1-axis x = np.array([3, 2]) print(F @ (R @ x), (F @ R) @ x) print(R @ (F @ x), (R @ F) @ x) print(F @ R) print(R @ F) print(np.allclose(rotation(30) @ rotation(60), rotation(90)), np.allclose(rotation(60) @ rotation(30), rotation(90))) print(R @ R) H = np.array([[1, 1], [0, 1]]) S = np.array([[2, 0], [0, 0.5]]) print(H @ H @ H) print(S @ H) print(H @ S)

⌘/Ctrl + Enter runs the cell

Lesson 3.2.3 asks which maps can be undone, and finds the matrix that undoes them. Module 3.6 looks for the directions that a map only stretches.

Common mistakes

Columns, not rows. Column $j$ of $A$ is where $\mathbf{e}_j$ goes. Writing the images as rows gives a different matrix, and a different map.
$AB$ means $B$ first. To rotate and then reflect, use $FR$ , not $RF$ .
A shift is not linear. A linear map always keeps the origin where it is, so no $2 \times 2$ matrix can shift the plane.
Mind the sign in a rotation. For an anticlockwise turn, $-\sin\theta$ sits in row 1, column 2. Moving the minus sign to row 2 turns the other way.
Radians in code. np.cos and np.sin take radians; convert degrees with np.radians.
A projection loses information. Many points land on the same image, so you cannot tell which one you started from.

Matrices as maps

0 of 12 correct

1.
A linear map sends $\mathbf{u}$ to $(2, 1)$ and $\mathbf{v}$ to $(-1, 4)$ . Find the second entry of the image of $\mathbf{u} + \mathbf{v}$
2.
The same map: find the first entry of the image of $3\mathbf{u} - \mathbf{v}$
3.
For the shift $T(\mathbf{x}) = \mathbf{x} + (1, 1)$ , find the first entry of $T(\mathbf{u} + \mathbf{v}) - T(\mathbf{u}) - T(\mathbf{v})$ , for any $\mathbf{u}$ and $\mathbf{v}$
4.
A linear map sends $\mathbf{e}_1$ to $(3, -1)$ and $\mathbf{e}_2$ to $(2, 5)$ . What is the entry $a_{12}$ of its matrix?
5.
The shear $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ sends $(1, 3)$ where? Give the first entry
6.
Rotate $(4, 1)$ a quarter turn anticlockwise. What is the first entry of the image?
7.
Rotate $(2, 0)$ through 60° anticlockwise. What is the second entry of the image? (type sqrt for a square root)
8.
Reflect $(3, 5)$ in the line $x_2 = x_1$ . What is the first entry of the image?
9.
Under the projection onto the $x_1$ -axis, how many of the points $(2, 0)$ , $(2, 5)$ , $(-2, 5)$ and $(2, -1)$ land on $(2, 0)$ ?
10.
Rotate a quarter turn anticlockwise, then reflect in the $x_1$ -axis. Where does $(1, 0)$ go? Give the second entry
11.
Now reflect first, then rotate a quarter turn. Where does $(1, 0)$ go? Give the second entry
12.
For the shear $H = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ , find the top-right entry of $HHH$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

A

Matrices as linear maps

A matrix moves the plane

What makes a map linear

Rule: A linear map

The columns are where e₁ and e₂ go

Rule: The columns are the images of the basis

A gallery of maps

Rule: Rotation through θ

One map after another

Rule: One map after another

Common mistakes

Matrices as maps

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