MATH 133 Linear Algebra and Geometry • McGill University, Montreal

Revision sheet: linear transformations and their matrices (MATH 133)

Q: How do I find the standard matrix of a linear transformation?

Compute the image of each standard basis vector, first e1, then e2, and so on, and write those images as the columns of the matrix, in order. This works whether the map is given by a formula, by a picture such as a rotation or a reflection, or by its values on other vectors, which you first combine to recover the images of the basis vectors.

Q: How do I show that a transformation is not linear?

First check where the zero vector goes: if it does not go to zero, the map is not linear and you are done. Otherwise find one explicit pair of vectors whose sum is not sent to the sum of their images, or one vector and one scalar that break the scaling rule, and write the numbers on both sides. A single numerical counterexample is a complete proof.

Q: Why is the order reversed when you compose two linear transformations?

Because the matrix that touches the vector acts first. If T is applied first and S second, the result is S of T of x, which is the matrix of S times the matrix of T times x. So the map you apply first is written on the right. Check by following the first basis vector through both maps by hand and comparing with the first column of your product.

Q: Does a projection onto a line have an inverse?

No. A projection sends every point of a line perpendicular to the target line to the same point, so the original point cannot be recovered. Its matrix has determinant zero and the unit square is crushed onto a segment of area zero. Rotations are undone by rotating back, reflections are undone by reflecting again, but a projection throws information away for good.

Q: How can I tell from the rank whether a linear map is one-to-one or onto?

Count the pivots of the matrix to get the rank. The map is one-to-one exactly when the rank equals the number of columns, the dimension of the starting space, and onto exactly when the rank equals the number of rows, the dimension of the target space. A map into a smaller space can never be one-to-one, and a map into a bigger space can never be onto.

This sheet is not a summary of Nicholson sections 2.6 and 4.4: you already have the notes. It answers one question only, what makes students lose marks on linear transformations in MATH 133 at McGill University, and which precise gesture avoids each loss.

Almost every trap below is the same mistake in disguise: guessing the matrix instead of computing where $e_1$ and $e_2$ go. Train that single reflex, the images of the standard basis written as columns, and the rotation, the mirror, the projection, the composition and the kernel all become the same exercise.

The thread of the chapter

A linear transformation is decided by where it sends $e_1, \dots, e_n$ , and those images ARE the columns of its matrix: to find a matrix, ask where the standard basis goes; to read one, read its columns; to compose, apply the right-hand factor first.

This chapter is part of MATH 133, Linear Algebra and Geometry (McGill)

What is on this sheet

1. The essentials
2. The mistakes that cost marks
3. Which method to choose
4. How the answer is expected to be written
5. Check before you hand in
6. The typical problem, taken apart
7. Learn by heart
8. Frequently asked questions

Topics covered

Linear transformation
Standard matrix
Matrix transformation
Rotation matrix
Reflection across a line
Orthogonal projection onto a line
Shear and stretch
Composition of linear transformations
Inverse transformation
Area scaling factor
Linear operators on three-dimensional space
Kernel and image

The essentials

The columns are the images of the standard basis

• $T : \mathbb{R}^n \to \mathbb{R}^m$ is linear when $T(u + v) = T(u) + T(v)$ and $T(cu) = cT(u)$ for all $u, v, c$ . Then $T(0) = 0$ , but $T(0) = 0$ alone proves nothing.
• Every linear $T$ is $T(x) = Ax$ with $A = \begin{pmatrix} T(e_1) & \cdots & T(e_n) \end{pmatrix}$ : the $j$ -th COLUMN is $T(e_j)$ , and $A$ is $m \times n$ .
• Reading backwards: $Ax = x_1 T(e_1) + \cdots + x_n T(e_n)$ , so the image of $T$ is the span of the columns.
• Composition: $S \circ T$ ( $T$ first) has matrix $A_S A_T$ . The first map applied is the factor next to $x$ .
• $T$ multiplies every area by $|\det A|$ ; $\det A < 0$ means orientation is reversed, $\det A = 0$ means the plane is crushed.

For

A = \begin{pmatrix} 2 & -1 \\ 1 & 1 \end{pmatrix}

: the orange arrow is column 1, the image of

e_1

; the green arrow is column 2, the image of

e_2

; the dashed parallelogram is the image of the square.

On a midterm, writing “ $T(e_1) = \dots$ , $T(e_2) = \dots$ , so $A = \dots$ ” before the matrix earns the method mark even when an entry is wrong.

The standard matrices of the plane, and where they come from

• Rotation by $\theta$ counterclockwise: $R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ , because $e_1 \mapsto (\cos\theta, \sin\theta)$ and $e_2 \mapsto (-\sin\theta, \cos\theta)$ .
• Projection onto $y = mx$ : $P = \frac{1}{1+m^2}\begin{pmatrix} 1 & m \\ m & m^2 \end{pmatrix}$ , from $\text{proj}_d(e_i)$ with $d = (1, m)$ .
• Reflection across $y = mx$ : $F = 2P - I = \frac{1}{1+m^2}\begin{pmatrix} 1-m^2 & 2m \\ 2m & m^2-1 \end{pmatrix}$ , because $P(v)$ is the midpoint of $v$ and $F(v)$ .
• Horizontal shear $(x, y) \mapsto (x + ky, y)$ : $\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$ . Stretch: $\begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}$ .
• In $\mathbb{R}^3$ : projection onto the plane of normal $n$ is $I - \frac{nn^{T}}{n \cdot n}$ , reflection through it is $I - \frac{2nn^{T}}{n \cdot n}$ ; a rotation about a coordinate axis keeps that axis and turns the other two.

The mistakes that cost marks

These are the errors I correct most often in session. Each one costs marks on a paper, even when the reasoning behind it is right.

1. Writing the images of the basis as rows instead of columns

the whole question, and every part that uses the matrix

What not to write

“ $T(x, y) = (x + 2y, 3y)$ , so $T(e_1) = (1, 0)$ and $T(e_2) = (2, 3)$ , and $A = \begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix}$ .”

What to write

“ $T(e_1) = (1, 0)$ and $T(e_2) = (2, 3)$ are the COLUMNS: $A = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ , and $A(0, 1) = (2, 3)$ as required.”

Why: $Ae_j$ picks out the $j$ -th column of $A$ , so the column is where $T(e_j)$ must sit. The row version is the transpose: it sends $e_2$ to $(0, 3)$ , and one multiplication by $e_2$ exposes it. For a non-square map the row version does not even have the right size.

2. Multiplying a composition in reading order

the whole question, with no warning from the arithmetic

What not to write

“Rotate by $\pi/2$ , then project onto the $x$ -axis: the matrix is $RP$ , since $R$ comes first.”

What to write

“ $R$ acts first, so it sits next to $x$ : the matrix is $PR = \begin{pmatrix} 0 & -1 \\ 0 & 0 \end{pmatrix}$ , and $PR(2, 1) = (-1, 0)$ .”

From the same

v = (2, 1)

: rotating first then projecting lands on

(-1, 0)

, orange; projecting first then rotating lands on

(0, 2)

, green. Two orders, two answers.

Why: $(P \circ R)(x) = P(Rx)$ , and the matrix touching $x$ acts first. In the other order the result is $RP(2, 1) = (0, 2)$ , a different point on a different axis. Test the first column by hand: $R(e_1) = e_2$ , then $P(e_2) = 0$ , so column 1 of the right answer is $0$ .

3. Putting the minus sign of the rotation in the wrong place

2 marks, and every angle after it turns the wrong way

What not to write

“Rotation by $\theta$ counterclockwise: $\begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}$ .”

What to write

“ $e_1 \mapsto (\cos\theta, \sin\theta)$ and $e_2 \mapsto (-\sin\theta, \cos\theta)$ , so $R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ .”

Why: The wrong matrix is the rotation by $-\theta$ , clockwise. Test with $\theta = \pi/2$ : the correct matrix must send $e_1$ to $e_2 = (0, 1)$ , the wrong one sends it to $(0, -1)$ . Never memorize the sign pattern; derive it from where $e_1$ goes, in five seconds.

4. Trusting the origin test as a proof of linearity

the whole part

What not to write

“ $Q(x, y) = (x^2, y)$ satisfies $Q(0, 0) = (0, 0)$ , so $Q$ is linear.”

What to write

“ $Q(0) = 0$ is necessary but not sufficient: $Q(2, 0) = (4, 0) \neq 2Q(1, 0) = (2, 0)$ , so $Q$ is not linear.”

Why: The origin test only KILLS candidates, such as translations and $f(x) = 3x + 2$ , which high school calls linear and linear algebra calls affine. A proof of linearity needs letters for both rules; a disproof needs one set of numbers.

5. Using the images of non-standard vectors directly as columns

the whole question

What not to write

“ $T(1, 2) = (5, 0)$ and $T(0, 1) = (1, -1)$ , so $A = \begin{pmatrix} 5 & 1 \\ 0 & -1 \end{pmatrix}$ .”

What to write

“ $e_1 = (1, 2) - 2(0, 1)$ , so $T(e_1) = (5, 0) - 2(1, -1) = (3, 2)$ and $A = \begin{pmatrix} 3 & 1 \\ 2 & -1 \end{pmatrix}$ .”

Why: The columns are the images of $e_1$ and $e_2$ and of nothing else. Recover them by writing each $e_i$ as a combination of the given vectors, or all at once with $A = WV^{-1}$ . Then check: $A(1, 2) = (3 + 2, 2 - 2) = (5, 0)$ .

6. Forgetting to normalize the direction vector in a reflection

2 marks, and an answer that stretches the plane

What not to write

“Reflection across $y = 3x$ with $d = (1, 3)$ : $F = 2dd^{T} - I = \begin{pmatrix} 1 & 6 \\ 6 & 17 \end{pmatrix}$ .”

What to write

“ $F = \frac{2dd^{T}}{d \cdot d} - I = \frac{1}{5}\begin{pmatrix} -4 & 3 \\ 3 & 4 \end{pmatrix}$ , whose columns have length $1$ .”

Why: The projection formula divides by $d \cdot d = 1 + m^2 = 10$ . A mirror never changes lengths, so each column of a reflection matrix must have length $1$ : the wrong column $(1, 6)$ has length $\sqrt{37}$ , which settles it before any other check.

7. Treating a projection as an invertible map

the whole part, often a full question on invertibility

What not to write

“To recover $v$ from its projection onto $L$ , apply $P^{-1}$ .”

What to write

“ $\det P = 0$ : $P$ has no inverse, since every point of a line perpendicular to $L$ has the same projection.”

Why: For $P = \frac{1}{5}\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$ , both $(5, 0)$ and $(1, 2)$ go to $(1, 2)$ : no map can send $(1, 2)$ back to both. Undo a rotation with $R_{-\theta}$ , undo a reflection with itself, but a projection throws information away and cannot be undone.

8. Copying the sine pattern of the z-axis onto the y-axis

2 marks, and a rotation in the wrong direction

What not to write

“Rotation about the $y$ -axis: $\begin{pmatrix} \cos\theta & 0 & -\sin\theta \\ 0 & 1 & 0 \\ \sin\theta & 0 & \cos\theta \end{pmatrix}$ .”

What to write

“About the $y$ -axis the turn goes from $z$ to $x$ : $e_3 \mapsto (\sin\theta, 0, \cos\theta)$ , so the matrix is $\begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{pmatrix}$ .”

Why: The cyclic order is $x \to y \to z \to x$ , so for the $y$ -axis the plane turns from $z$ towards $x$ , and the minus sign moves to the bottom row. Test with $\theta = \pi/2$ : the correct matrix sends $e_3$ to $e_1$ .

Which method to choose

Which route to the standard matrix, from what the statement gives

Look at the FORM of the data, not at the story around it

If a formula $T(x, y, \dots) = (\dots)$ → compute $T(e_1), \dots, T(e_n)$ and stand them up as columns; the rows must match the coefficients of the formula

Example: $T(x, y, z) = (x - 2y + 4z, 3x + z)$ gives $\begin{pmatrix} 1 & -2 & 4 \\ 3 & 0 & 1 \end{pmatrix}$
If a rotation, a shear or a stretch described in words → draw $e_1$ , $e_2$ and where they land; read the coordinates with trigonometry

Example: shear by $3$ : $e_1$ stays, $e_2 \mapsto (3, 1)$ , so $\begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}$
If a line $y = mx$ and the words project or reflect → project $e_1$ and $e_2$ on $d = (1, m)$ , then $F = 2P - I$

Example: $y = 2x$ : $P = \frac{1}{5}\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$ , $F = \frac{1}{5}\begin{pmatrix} -3 & 4 \\ 4 & 3 \end{pmatrix}$
If a plane $n \cdot x = 0$ in $\mathbb{R}^3$ → $P(e_i) = e_i - \frac{e_i \cdot n}{n \cdot n}n$ for each $i$ , then $Q = 2P - I$ for the reflection

Example: $n = (1, 1, 1)$ : $P(e_1) = \frac{1}{3}(2, -1, -1)$
If images of vectors that are NOT $e_1, \dots, e_n$ → write each $e_i$ as a combination of the given vectors, or use $A = WV^{-1}$

Example: $T(1, 1) = (3, 1)$ , $T(1, -1) = (1, 5)$ give $\begin{pmatrix} 2 & 1 \\ 3 & -2 \end{pmatrix}$
If the words first, then, followed by, after → multiply the matrices from right to left: the first map applied goes on the right

Example: rotate by $\pi/2$ then reflect across the $x$ -axis: $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$

No branch applies? Then the statement always lets you compute $T(e_1)$ somehow, and that is the whole method. Changing to a non-standard basis is outside MATH 133: every matrix you write here is the standard one.

Recognizing a 2 by 2 matrix by its form

Look at the columns and the determinant before computing anything

Where

e_1

(orange) and

e_2

(green) land: turned together, swapped by the mirror

y = x

e_2

crushed to

0

by the projection,

e_2

slid to

(1, 1)

by the shear.

If columns of length $1$ , perpendicular, $\det = 1$ → a rotation, by the angle of the first column

Example: $\frac{1}{5}\begin{pmatrix} 3 & -4 \\ 4 & 3 \end{pmatrix}$ : $\cos\theta = 3/5$ , $\sin\theta = 4/5$
If columns of length $1$ , perpendicular, $\det = -1$ → a reflection across a line; it is its own inverse

Example: $\frac{1}{5}\begin{pmatrix} -3 & 4 \\ 4 & 3 \end{pmatrix}$ , the mirror $y = 2x$
If $\det = 0$ → the plane is crushed onto a line or a point: no inverse; a projection if moreover $A^2 = A$

Example: $\frac{1}{5}\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$ , the projection onto $y = 2x$
If triangular with $1$ on the diagonal → a shear; areas are kept since $\det = 1$

Example: $\begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}$ sends $e_2$ to $(3, 1)$
If diagonal → a stretch along each axis, areas multiplied by the product of the diagonal

Example: $\begin{pmatrix} 2 & 0 \\ 0 & 1/3 \end{pmatrix}$ multiplies areas by $2/3$

Perpendicular columns of equal length $k \neq 1$ mean a rotation or a reflection followed by a dilation by $k$ : $\begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix} = \sqrt{2}\,R_{-\pi/4}$ .

One-to-one, onto, invertible: read it on the rank

$T : \mathbb{R}^n \to \mathbb{R}^m$ with matrix $A$ of rank $r$ , counted in pivots

If $r = n$ , the number of COLUMNS → one-to-one: $\ker T = \{0\}$

Example: $(x, y) \mapsto (x, y, 0)$ has $r = 2 = n$
If $r = m$ , the number of ROWS → onto: the columns span $\mathbb{R}^m$

Example: the sensor $\mathbb{R}^4 \to \mathbb{R}^3$ of rank $3$ is onto
If $m = n = r$ → invertible, and $T^{-1}$ has matrix $A^{-1}$

Example: $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ , $\det = 3 \neq 0$
If $n > m$ → never one-to-one, since $r \le m < n$

Example: any map $\mathbb{R}^3 \to \mathbb{R}^2$ has a non-zero kernel

How the answer is expected to be written

A marker ticks steps. Here they are in order, with the concluding sentence expected word for word.

Finding the standard matrix of a transformation

When to use it: Any question that says find the matrix of, the standard matrix, or the matrix that represents

1 Say why $T$ is linear, or quote that the transformation is a rotation, reflection or projection, known to be linear.
2 Compute $T(e_1), \dots, T(e_n)$ one by one, each on its own line, with the picture or the formula that gives it.
3 Write them as the COLUMNS of $A$ , and state the size $m \times n$ .
4 Test $A$ on one vector whose image you know independently: a point on the mirror, the normal of a plane, a given data vector.
5 Conclude with the sentence below.

Concluding sentence

“Since $T$ is linear, $T(x) = Ax$ where $A = \begin{pmatrix} T(e_1) & T(e_2) \end{pmatrix}$ is the standard matrix of $T$ , whose columns are the images of the standard basis vectors.”

The trap: Writing the matrix straight from memory with no images: if one sign is wrong, there is nothing on the page to give a method mark for.

Marking: Typically 1 mark per correct image of a basis vector, 1 for the matrix assembled with the images as columns, and 1 for a check or the concluding sentence.

Proving or disproving linearity

When to use it: Is T linear? Justify. Or: show that T is a linear transformation

1 Compute $T(0)$ first. If it is not $0$ , stop: that line alone disproves linearity.
2 To DISPROVE otherwise: find one explicit $u, v$ with $T(u + v) \neq T(u) + T(v)$ , or one $u, c$ with $T(cu) \neq cT(u)$ , and write the numbers on both sides.
3 To PROVE: take arbitrary $u = (x_1, y_1)$ , $v = (x_2, y_2)$ and a scalar $c$ , and verify both rules with letters, regrouping coordinate by coordinate.
4 Alternatively, exhibit a matrix $A$ with $T(x) = Ax$ for all $x$ : a matrix transformation is always linear.

Concluding sentence

“For all $u, v \in \mathbb{R}^2$ and all $c \in \mathbb{R}$ , $T(u + v) = T(u) + T(v)$ and $T(cu) = cT(u)$ , hence $T$ is linear.”

The trap: Checking the two rules on a particular pair of vectors and concluding that T is linear: an example never proves a statement about all vectors.

Check before you hand in

Five minutes of checking recover more marks than one more problem started in a hurry.

The size test

A map from $\mathbb{R}^n$ to $\mathbb{R}^m$ has an $m \times n$ matrix: one column per INPUT dimension. If the product $Ax$ cannot be formed with $x$ in the domain, the images went into rows.

$T : \mathbb{R}^3 \to \mathbb{R}^2$ with a $3 \times 2$ matrix is wrong on sight; it must be $2 \times 3$ .
Columns of a rotation or a reflection

Both keep lengths, so each column has length $1$ and the two columns are perpendicular. Then $\det = 1$ for a rotation, $-1$ for a reflection, $0$ for a projection.

$\frac{1}{5}(-4, 3)$ has length $1$ : plausible. $(1, 6)$ has length $\sqrt{37}$ : a factor $\frac{1}{1+m^2}$ is missing.
One vector you know without the matrix

Apply your matrix to a vector whose image the geometry gives for free: a point on the mirror does not move, the normal of a projection plane goes to 0, the midpoint of a point and its reflection lies on the mirror.

For $F = \frac{1}{5}\begin{pmatrix} -3 & 4 \\ 4 & 3 \end{pmatrix}$ : $F(5, 0) = (-3, 4)$ and the midpoint $(1, 2)$ lies on $y = 2x$ .
The first column of a composition

Follow $e_1$ through the maps by hand, in the order they act, and compare with column 1 of your product.

Rotate by $\pi/2$ , then project onto the $x$ -axis: $e_1 \mapsto e_2 \mapsto 0$ , so column 1 must be $0$ . $RP$ has column 1 equal to $(0, 1)$ : wrong order.
The area check

The image of the unit square has area $|\det A|$ . A shear or a rotation keeps it at $1$ , a reflection too, a projection sends it to $0$ .

$\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ sends the square onto a parallelogram of area $3$ , so a disc of area $4\pi$ becomes $12\pi$ .

The typical problem, taken apart

Reflect across y = 3x, then turn by a quarter turn

Let $L$ be the line $y = 3x$ . Find the standard matrices of the projection $P$ onto $L$ and of the reflection $F$ across $L$ .

Then find the matrix of the transformation that reflects across $L$ and THEN rotates by $\pi/2$ counterclockwise, and identify it geometrically.

P(e_1)

is the foot of the perpendicular from

e_1

L

, and the midpoint between

e_1

and

F(e_1)

: that is why

F = 2P - I

Step 1

$d = (1, 3)$ , $d \cdot d = 10$ . $P(e_1) = \frac{1}{10}(1, 3)$ and $P(e_2) = \frac{3}{10}(1, 3)$ , so $P = \frac{1}{10}\begin{pmatrix} 1 & 3 \\ 3 & 9 \end{pmatrix}$ .

Why

The images of $e_1$ and $e_2$ come first, each written out: that is where the method marks are, and the division by $d \cdot d$ is visible instead of forgotten.

Step 2

$F = 2P - I = \frac{1}{10}\begin{pmatrix} 2 - 10 & 6 \\ 6 & 18 - 10 \end{pmatrix} = \frac{1}{5}\begin{pmatrix} -4 & 3 \\ 3 & 4 \end{pmatrix}$ .

Why

Deriving $F$ from $P$ with the midpoint argument avoids memorizing a second formula, and it works unchanged for a plane in $\mathbb{R}^3$ .

Step 3

Check: $F(1, 3) = \frac{1}{5}(-4 + 9,\ 3 + 12) = (1, 3)$ , and the column $\frac{1}{5}(-4, 3)$ has length $1$ .

Why

A point of the mirror must not move, and a mirror keeps lengths: two checks that come from the geometry, not from the formula, so they catch a wrong formula.

Step 4

The reflection acts first, so it goes on the right: $M = R_{\pi/2}F = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\frac{1}{5}\begin{pmatrix} -4 & 3 \\ 3 & 4 \end{pmatrix} = \frac{1}{5}\begin{pmatrix} -3 & -4 \\ -4 & 3 \end{pmatrix}$ .

Why

Right to left is the rule of composition. Column 1 checks it by hand: $F(e_1) = \frac{1}{5}(-4, 3)$ , turned by a quarter turn, is $\frac{1}{5}(-3, -4)$ .

Step 5

$\det M = \frac{1}{25}(-9 - 16) = -1$ and the columns are perpendicular of length $1$ : $M$ is a reflection. Its mirror contains $(1, -2)$ , since $M(1, -2) = \frac{1}{5}(-3 + 8,\ -4 - 6) = (1, -2)$ : the line $y = -2x$ .

Why

The recognition tree decides on the form: orthonormal columns with $\det = -1$ means a reflection. A reflection followed by a rotation is again a reflection, since $(-1)(1) = -1$ .

The conclusion, written out

“ $P = \frac{1}{10}\begin{pmatrix} 1 & 3 \\ 3 & 9 \end{pmatrix}$ , $F = \frac{1}{5}\begin{pmatrix} -4 & 3 \\ 3 & 4 \end{pmatrix}$ , and reflecting across $L$ then rotating by $\pi/2$ has matrix $\frac{1}{5}\begin{pmatrix} -3 & -4 \\ -4 & 3 \end{pmatrix}$ : the reflection across the line $y = -2x$ .”

The classic mistake on this problem: Multiplying in reading order, $FR_{\pi/2} = \frac{1}{5}\begin{pmatrix} 3 & 4 \\ 4 & -3 \end{pmatrix}$ : also a reflection, but across $y = x/2$ , since it fixes $(2, 1)$ . The answer looks just as clean and is worth nothing.

Learn by heart

• The $j$ -th COLUMN of the standard matrix is $T(e_j)$ ; a map $\mathbb{R}^n \to \mathbb{R}^m$ has an $m \times n$ matrix.
• $T(0) = 0$ is necessary, never sufficient. Affine maps such as $x \mapsto 3x + 2$ are NOT linear.
• $R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ counterclockwise; $R_\theta^{-1} = R_{-\theta} = R_\theta^{T}$ .
• Line $y = mx$ : $P = \frac{1}{1+m^2}\begin{pmatrix} 1 & m \\ m & m^2 \end{pmatrix}$ , $F = 2P - I$ , $F^{-1} = F$ , $P$ has no inverse.
• The matrix of $S \circ T$ is $A_S A_T$ : the map applied FIRST sits on the RIGHT.
• Areas are multiplied by $|\det A|$ ; $\det < 0$ reverses orientation.
• One-to-one iff $\ker T = \{0\}$ iff rank $= n$ ; onto iff rank $= m$ .

Frequently asked questions

How do I find the standard matrix of a linear transformation?

Compute the image of each standard basis vector, first e1, then e2, and so on, and write those images as the columns of the matrix, in order. This works whether the map is given by a formula, by a picture such as a rotation or a reflection, or by its values on other vectors, which you first combine to recover the images of the basis vectors.

How do I show that a transformation is not linear?

First check where the zero vector goes: if it does not go to zero, the map is not linear and you are done. Otherwise find one explicit pair of vectors whose sum is not sent to the sum of their images, or one vector and one scalar that break the scaling rule, and write the numbers on both sides. A single numerical counterexample is a complete proof.

Why is the order reversed when you compose two linear transformations?

Because the matrix that touches the vector acts first. If T is applied first and S second, the result is S of T of x, which is the matrix of S times the matrix of T times x. So the map you apply first is written on the right. Check by following the first basis vector through both maps by hand and comparing with the first column of your product.

Does a projection onto a line have an inverse?

No. A projection sends every point of a line perpendicular to the target line to the same point, so the original point cannot be recovered. Its matrix has determinant zero and the unit square is crushed onto a segment of area zero. Rotations are undone by rotating back, reflections are undone by reflecting again, but a projection throws information away for good.

How can I tell from the rank whether a linear map is one-to-one or onto?

Count the pivots of the matrix to get the rank. The map is one-to-one exactly when the rank equals the number of columns, the dimension of the starting space, and onto exactly when the rank equals the number of rows, the dimension of the target space. A map into a smaller space can never be one-to-one, and a map into a bigger space can never be onto.

Practise it

Corrected exercises: Linear transformations and their matrices, MATH 133 at McGill

A method is proved on a paper, not on a sheet. The set for the same chapter takes each of these traps into a problem, with the solution written out step by step.

10 corrected exercises
100 points
150 minutes

Do the exercises

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Get in touch for a first session. Linear transformations are where the course turns geometric: one reflex, the images of the basis as columns, and every chapter after this one gets easier.

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Revision sheet: linear transformations and their matrices (MATH 133)

The essentials

The columns are the images of the standard basis

The standard matrices of the plane, and where they come from

The mistakes that cost marks

1. Writing the images of the basis as rows instead of columns

2. Multiplying a composition in reading order

3. Putting the minus sign of the rotation in the wrong place

4. Trusting the origin test as a proof of linearity

5. Using the images of non-standard vectors directly as columns

6. Forgetting to normalize the direction vector in a reflection

7. Treating a projection as an invertible map

8. Copying the sine pattern of the z-axis onto the y-axis

Which method to choose

Which route to the standard matrix, from what the statement gives

Recognizing a 2 by 2 matrix by its form

One-to-one, onto, invertible: read it on the rank

How the answer is expected to be written

Finding the standard matrix of a transformation

Proving or disproving linearity

Check before you hand in

The typical problem, taken apart

Reflect across y = 3x, then turn by a quarter turn

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Frequently asked questions

How do I find the standard matrix of a linear transformation?

How do I show that a transformation is not linear?

Why is the order reversed when you compose two linear transformations?

Does a projection onto a line have an inverse?

How can I tell from the rank whether a linear map is one-to-one or onto?

Corrected exercises: Linear transformations and their matrices, MATH 133 at McGill

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