Linear Algebra and Vector Geometry 201-SN4-RE • CEGEP, Montreal

Revision sheet: linear transformations of the plane (201-SN4-RE)

Q: How do you find the matrix of a transformation of the plane?

Compute the image of the vector one, zero and the image of the vector zero, one, then put these two images in columns. The first image becomes the first column, the second image the second column. This rule holds only for the vectors of the standard basis, not for any two vectors.

Q: In which order do you multiply the matrices of a composition?

The transformation applied first goes on the right of the product. If you rotate and then reflect, the matrix is the reflection matrix multiplied by the rotation matrix. The reason is that the right-hand factor is the one that meets the vector first in the calculation.

Q: Why does the reflection matrix use the double angle?

Because a point at a given angle is sent to the other side of the axis, at the same angular distance. Its new angle is therefore twice the angle of the axis minus its starting angle. It is the gap that doubles, whereas a rotation simply adds its angle.

Q: What does a zero determinant mean for a transformation?

That the transformation flattens the plane onto a line or onto a point. It does exist, but it is not invertible: several different points have the same image, and the area of every transformed figure becomes zero. Projections are the typical example of this case.

Q: How do you recognize a rotation from its matrix?

Its determinant is one, and its two columns have length one and are perpendicular to each other. The angle can then be read directly from the first column, whose coordinates are the cosine and the sine of that angle. A determinant of minus one would signal a reflection, not a rotation.

Q: Is a translation a linear transformation?

No. A linear transformation always leaves the origin fixed, whereas a translation moves it. A translation is called affine: it is a linear transformation followed by the addition of a constant vector, and only the linear part can be represented by a matrix.

This sheet does not repeat the list of rotation and reflection matrices: it is in your notes and on the formula sheet. It deals with what decides the mark, that is, the order of a composition, reading the columns, and the checks that catch a wrong matrix in ten seconds.

It is written for CEGEP students in the Science program taking Linear Algebra and Vector Geometry, 201-SN4-RE (formerly 201-NYC). Once you have read it, the corrected exercise set on the same chapter puts each reflex to the test.

Mark this sheet as read or add it to your favourites: a free account, no password, keeps your read sheets and favourites from one visit to the next, tells you which chapter to tackle next and lets you ask for the method or exercise you are missing. Create your space, an email is enough.

The thread of the chapter

The matrix of a transformation is READ from the images of $(1, 0)$ and $(0, 1)$ : they are its two columns. And a composition is read from RIGHT to left, because the right-hand factor is the one that meets the vector first.

This chapter is part of 201-SN4-RE, Linear Algebra and Vector Geometry

The essentials

The columns are the images of the standard basis

• Every linear transformation of the plane can be written $T(\vec{v})=A\vec{v}$ for a UNIQUE $2\times 2$ matrix $A$ .
• The first column of $A$ is $T(1, 0)$ , the second is $T(0, 1)$ . This is the key in both directions: from the geometry to the matrix, and from the matrix to the geometry.
• $T$ is linear if and only if $T(\vec{u}+\vec{v})=T(\vec{u})+T(\vec{v})$ and $T(k\vec{u})=k\,T(\vec{u})$ . In particular, $T(\vec{0})=\vec{0}$ always.
• The image of the UNIT SQUARE is the parallelogram built on the two columns: the fastest way to see what a matrix does.
• A translation is never linear: it moves the origin.

The unit square (blue) becomes the parallelogram (orange) built on

T(\vec{e_{1}})

and

T(\vec{e_{2}})

, exactly the two columns of

\begin{pmatrix} 2 & 1 \\ 0.5 & 1.5 \end{pmatrix}

Faced with an unknown matrix, sketching the image of the unit square takes thirty seconds and shows at once whether it is a rotation, a reflection, a shear or a flattening.

The catalogue, with the determinants that identify

• ROTATION through $\theta$ : $R_{\theta}=\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ , with determinant $1$ . The minus sign is at the TOP right.
• REFLECTION about the $x$ -axis: $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ ; about the $y$ -axis: $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ ; about $y=x$ : $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ . Determinant $-1$ .
• REFLECTION about the line at angle $\theta$ : $\begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}$ , with the DOUBLE angle.
• DILATION with factor $k$ : $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$ , with determinant $k^{2}$ . PROJECTION onto the $x$ -axis: $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ , with determinant $0$ .
• SHEAR: $\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$ , with determinant $1$ : it preserves area without being a rotation.

R_{\theta}

sends

\vec{e_{1}}

(\cos\theta, \sin\theta)

and

\vec{e_{2}}

(-\sin\theta, \cos\theta)

, its two columns:

\vec{e_{2}}

swings to the LEFT, hence the minus sign at the top right.

The determinant identifies almost everything: $1$ for a rotation or a shear, $-1$ for a reflection, $0$ for a projection. A matrix is a rotation if, in addition, its columns are unit vectors and orthogonal.

Composing, and reading the area factor

• The matrix of $T_{2}\circ T_{1}$ is the PRODUCT $A_{2}A_{1}$ , in that order: the transformation applied FIRST goes on the RIGHT.
• The reason fits in one line: $T_{2}(T_{1}(\vec{v}))=A_{2}(A_{1}\vec{v})=(A_{2}A_{1})\vec{v}$ , and it is $A_{1}$ that touches $\vec{v}$ .
• Since matrix multiplication is not commutative, the order changes the result: a rotation followed by a reflection is not the same as the reflection followed by the rotation.
• If a figure of area $\mathcal{A}$ is transformed by $T$ , the image has area $|\det(A)|\times\mathcal{A}$ .
• $\det(A)<0$ signals a reversal of ORIENTATION; $\det(A)=0$ signals that the whole plane is flattened onto a line.

Left: a matrix with determinant

3

sends the square of area

1

onto an area of

3

. Right: the reflection about the

x

-axis (

\det=-1

) keeps the area but moves

\vec{e_{2}}

to the other side of

\vec{e_{1}}

Composing two reflections whose axes make an angle $\alpha$ always gives a ROTATION through $2\alpha$ : it is the most frequently asked result at the end of the chapter, and it checks out on the determinants, $(-1)\times(-1)=1$ .

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. Composing in reading order instead of matrix order

the whole question, with a matrix that is often the other reflection, and therefore perfectly plausible.

What not to write

“We first apply the rotation $R$ , then the reflection $S$ , so the matrix of the composition is $RS$ .”

What to write

“The matrix is $SR$ : the transformation applied FIRST goes on the RIGHT, because it is the one that meets the vector first in $S(R\vec{v})$ .”

The same L,

R

(rotation through

90^{\circ}

) and

S

(reflection about the

x

-axis): “first

R

” gives

SR

, the reflection about

y=-x

(middle); “first

S

” gives

RS

, about

y=x

(right).

Why: Matrix order is the reverse of chronological order. It is the only real difficulty of the chapter, and it comes from the fact that the notation $S\circ R$ is read from right to left.

2. Putting the minus sign on the wrong side of the rotation matrix

2 marks, and a rotation that turns the wrong way for the rest of the problem.

What not to write

“ $R_{\theta}=\begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}$ .”

What to write

“ $R_{\theta}=\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ : the minus sign is at the TOP right. Check: the first column is $R_{\theta}(1, 0)=(\cos\theta, \sin\theta)$ , which must go up for a positive $\theta$ .”

For

\theta>0

, the first column

R_{\theta}(1, 0)

must go UP, in green. The faulty matrix sends it below the axis, in red: that is the rotation through

-\theta

Why: The faulty matrix is that of $R_{-\theta}$ : it is consistent, so no later calculation contradicts it. Only the check on the first column exposes it.

3. Using the single angle for a reflection

the whole question, and the resulting matrix is not even a reflection about the right line.

What not to write

“The reflection about the line at angle $\theta$ has matrix $\begin{pmatrix} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{pmatrix}$ .”

What to write

“It is the DOUBLE angle: $\begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}$ . Immediate check: a vector LYING ON the line must stay unchanged.”

M

is at

10^{\circ}

, the axis

D

40^{\circ}

: the rotation through

40^{\circ}

sends it to

50^{\circ}

, the reflection about

D

70^{\circ}=2\times 40-10

, that is

30^{\circ}

on each side of the axis.

Why: A point at angle $\alpha$ is reflected to angle $2\theta-\alpha$ : it is the gap that doubles, not the angle of the line. A rotation, by contrast, simply adds $\theta$ .

4. Mixing up degrees and radians

the whole question, and the determinant is still

1

, so nothing flags the error.

What not to write

“Rotation through $60$ : $\cos 60=-0.952$ and $\sin 60=-0.305$ .”

What to write

“The calculator was in RADIANS. In degrees, $\cos 60^{\circ}=\frac{1}{2}$ and $\sin 60^{\circ}=\frac{\sqrt{3}}{2}$ , so $R_{60^{\circ}}=\begin{pmatrix} \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}$ .”

Why: The usual angles of the chapter, $30^{\circ}$ , $45^{\circ}$ , $60^{\circ}$ , $90^{\circ}$ , have exact values you must recognize. A negative cosine for an acute angle is the alarm bell.

5. Concluding that a transformation with zero determinant does not exist

2 marks, and the question on the kernel or the range becomes impossible to answer.

What not to write

“ $\det(A)=0$ , so this transformation does not exist.”

What to write

“It exists perfectly well, but it is not INVERTIBLE: it flattens the whole plane onto a line. Its range has dimension $1$ and its kernel is a whole line, not just $\{\vec{0}\}$ .”

The projection

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

does exist: it flattens the whole square onto a segment of the

x

-axis. The image area is zero, and all the points of a vertical line have the same image.

Why: A zero determinant says that the area factor is zero, so the image figure is flattened. That is exactly the case of a projection, the most useful transformation of the chapter.

6. Putting in columns the images of vectors that are not the basis vectors

the whole question, and the resulting matrix does not even satisfy the data of the problem.

What not to write

“ $T(1, 1)=(3, 2)$ and $T(1, -1)=(1, 4)$ , so $A=\begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$ .”

What to write

“The columns are the images of $(1, 0)$ and $(0, 1)$ . Since $(1, 0)=\frac{1}{2}[(1, 1)+(1, -1)]$ , linearity gives $T(1, 0)=\frac{1}{2}[(3, 2)+(1, 4)]=(2, 3)$ , and likewise $T(0, 1)=(1, -1)$ , so $A=\begin{pmatrix} 2 & 1 \\ 3 & -1 \end{pmatrix}$ .”

On the left,

\vec{e_{1}}

is half the sum of

(1, 1)

and

(1, -1)

. On the right, by linearity,

T(\vec{e_{1}})

is half the sum of their images,

(2, 3)

: that is what goes in the column.

Why: The column rule only holds for the STANDARD BASIS. With other vectors, first express $\vec{e_{1}}$ and $\vec{e_{2}}$ in terms of them, then use linearity.

7. Taking a translation for a linear transformation

the whole question, spent looking for a matrix that cannot exist.

What not to write

“ $T(x, y)=(x+1, y)$ is linear: it is a translation of the plane.”

What to write

“Every linear transformation sends $\vec{0}$ to $\vec{0}$ . Here $T(0, 0)=(1, 0)\neq\vec{0}$ : the translation is not linear, and it cannot be written $A\vec{v}$ .”

The translation

(x, y)\mapsto(x+1, y)

moves the origin itself, in red:

T(\vec{0})\neq\vec{0}

, so it is not linear and no

2\times 2

matrix represents it.

Why: A translation is an AFFINE transformation. It is written $\vec{v}\mapsto A\vec{v}+\vec{b}$ , and only the case $\vec{b}=\vec{0}$ is linear. The origin test settles it in one line.

8. Raising a matrix to a power entry by entry

the whole question, when the rotation shortcut would have given the answer in one line.

What not to write

“ $A^{3}$ is obtained by cubing each of the four entries of $A$ .”

What to write

“ $A^{3}=A\times A\times A$ , a MATRIX product. For a rotation there is a shortcut: $R_{\theta}^{n}=R_{n\theta}$ , so $R_{60^{\circ}}^{3}=R_{180^{\circ}}=-I$ .”

Why: The power of a matrix is a repeated composition, not an entry-by-entry calculation. For a rotation, composing means adding the angles, which saves the whole product.

Which method to choose

Which transformation, from the matrix alone

Compute the determinant, then look at the columns. Those two pieces of information are enough to identify every transformation in the course.

The unit square (dashed) and its image: the rotation keeps the shape, the shear tilts the square without changing its area, the reflection swaps the sides, the projection flattens it.

If $\det=1$ and the columns are orthogonal unit vectors → ROTATION, through the angle $\theta$ read on the first column $(\cos\theta, \sin\theta)$

Example: $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ is the rotation through $90^{\circ}$
If $\det=1$ but the columns are not unit vectors → SHEAR, or a transformation that preserves area without preserving lengths

Example: $\begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}$
If $\det=-1$ and the matrix is SYMMETRIC → REFLECTION, about the line of invariant vectors, found by solving $A\vec{v}=\vec{v}$

Example: $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ has axis $y=x$
If $\det=k^{2}$ with $A=kI$ → DILATION with factor $k$ , which multiplies every length by $|k|$
If $\det=0$ → the transformation FLATTENS the plane: range of dimension $1$ at most, kernel larger than $\{\vec{0}\}$

If, in addition, $A^{2}=A$ , it is a projection.

No branch applies? Sketch the image of the unit square: the two columns are enough to see what the matrix does, even when it has no name.

In which order to write the product

Find the word in the problem that gives the CHRONOLOGY, then reverse it to write the product.

If “first apply $T_{1}$ , then $T_{2}$ ” → the matrix is $A_{2}A_{1}$ , with the FIRST one on the right

Example: first the rotation $R$ , then the reflection $S$ , gives $SR$
If the problem writes $T_{2}\circ T_{1}$ → the matrix is $A_{2}A_{1}$ : the notation already has the right order

The small circle is read “composed with”, or “after”: $T_{2}\circ T_{1}$ is “ $T_{2}$ after $T_{1}$ ”.
If the problem describes an object transformed several times in a row → stack the matrices, adding each new transformation on the LEFT

Example: three steps $T_{1}$ , $T_{2}$ , $T_{3}$ give $A_{3}A_{2}A_{1}$
If you want to check the order you chose → apply the composition to a simple vector, $(1, 0)$ , and follow the two steps by hand

It is the fastest check, and it is never wrong.

The area factor, on the other hand, does not depend on the order: $|\det(A_{2}A_{1})|=|\det(A_{1}A_{2})|$ , since the determinant of a product is the product of the determinants.

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 matrix of a transformation described geometrically

When to use it: The problem describes a rotation, a reflection, a projection or a composition, and asks for its matrix.

1 State the rule: “the first column is the image of $(1, 0)$ , the second is the image of $(0, 1)$ ”.
2 Compute the image of $(1, 0)$ geometrically, with a sketch if the transformation is a reflection or a rotation.
3 Do the same for $(0, 1)$ , without assuming that the result follows from the first one.
4 Write the matrix by PUTTING these two images in columns, not in rows.
5 Check on a simple vector other than the basis vectors, and conclude with the determinant to identify the type of transformation.

Concluding sentence

The rotation through $90^{\circ}$ sends $(1, 0)$ to $(0, 1)$ and $(0, 1)$ to $(-1, 0)$ , so its matrix is $R=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ , with determinant $1$ .

The trap: Putting the images in ROWS. The matrix obtained is the transpose, that is, the inverse rotation: it is consistent, and therefore cannot be caught without testing it on a vector.

Marking: 0.5 mark for stating the rule, 0.5 mark per image computed, 0.5 mark for the matrix correctly arranged, 0.5 mark for the check.

Identifying a transformation from its matrix

When to use it: The problem gives a matrix and asks “which geometric transformation does it represent”.

1 Compute the DETERMINANT and state it: it points at once to a rotation, a reflection, a dilation or a flattening.
2 Check whether the columns are orthogonal unit vectors, which tells an isometry from a deformation.
3 Look for the INVARIANT vectors by solving $A\vec{v}=\vec{v}$ : a whole line of solutions signals a reflection or a projection.
4 Name the transformation with its defining element: the angle for a rotation, the axis for a reflection, the factor for a dilation.
5 Check on a well-chosen point that the description reproduces the action of the matrix.

Concluding sentence

$\det(A)=-1$ and $A$ is symmetric: it is a reflection; the invariant vectors satisfy $y=-x$ , so $A$ is the reflection about the line with equation $y=-x$ .

The trap: Stopping at “it is a reflection” without giving the AXIS. The question asks for the defining element, and it is worth half the marks of the question.

Marking: 0.5 mark for the determinant, 0.5 mark for the type, 1 mark for the defining element, 0.5 mark for the check.

Check before you hand in

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

Follow a simple vector through the composition

Take $(1, 0)$ , apply the transformations to it one at a time in chronological order, then compare with the matrix product you computed. The two must agree.

First $R$ then $S$ : $(1, 0)\to(0, 1)\to(0, -1)$ . The matrix $SR$ applied to $(1, 0)$ must give $(0, -1)$ , which confirms the order.
Check the determinant against the announced type

A rotation has determinant $1$ , a reflection $-1$ , a projection $0$ , a dilation with factor $k$ has $k^{2}$ . A disagreement exposes the matrix, not the name.

A matrix announced as a rotation with a determinant of $-1$ is in fact a reflection, and the error is almost always a misplaced sign.
Test that the axis of a reflection is invariant

A vector LYING ON the axis must stay unchanged, and a perpendicular vector must be sent to its opposite. Two ten-second matrix products validate the whole matrix.

For the reflection about $y=x$ , the matrix must fix $(1, 1)$ and send $(1, -1)$ to $(-1, 1)$ .
Check that the origin really is fixed

Every linear transformation satisfies $T(\vec{0})=\vec{0}$ . If the problem describes a transformation that moves the origin, it is not linear and no matrix represents it.

A rotation about the point $(2, 1)$ is not linear: you must first translate the origin, rotate, then translate back.
Compare the areas before and after

The area of the image is $|\det|$ times the original area. On the unit square, the image parallelogram must therefore have area exactly $|\det|$ , which can be checked on a sketch.

A shear with determinant $1$ produces a very elongated parallelogram with the same area as the square: the sketch confirms it, base and height unchanged.

The typical problem, taken apart

A rotation and a reflection, composed in both orders

Let $R$ be the rotation through $90^{\circ}$ about the origin and $S$ the reflection about the $x$ -axis.

1) Give the matrices of $R$ and $S$ .

2) Find the matrix of the transformation “first $R$ , then $S$ ”, and identify this transformation.

3) Do the same for “first $S$ , then $R$ ”, and compare.

Step 1

$R(1, 0)=(0, 1)$ and $R(0, 1)=(-1, 0)$ , so $R=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ ; likewise $S=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ .

Why

Read the images of the standard basis and put them IN COLUMNS. The minus sign of $R$ appears by itself, with no need to memorize the general formula.

Step 2

“First $R$ , then $S$ ” translates into $S\circ R$ , hence into the product $SR$ , with $R$ on the RIGHT.

Why

This is the only difficulty of the problem. Writing $S(R\vec{v})$ before setting up the product makes the order obvious, and is worth half a mark on most marking schemes.

Step 3

$SR=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}=\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$ .

Why

Each entry pairs a row of $S$ with a column of $R$ . Setting up the product in this order, and not the other, is what the previous step decided.

Step 4

$\det(SR)=0\times 0-(-1)\times(-1)=-1$ , and the matrix is symmetric: it is a REFLECTION.

Why

The determinant and the symmetry identify the type in one line. The axis is still to be found, or the answer is incomplete.

Step 5

Invariant vectors: $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}=\begin{pmatrix} x \\ y \end{pmatrix}$ gives $-y=x$ , so the axis is the line $y=-x$ .

Why

The axis of a reflection is exactly the set of vectors it does not move. Solving $A\vec{v}=\vec{v}$ is the general method, valid for every reflection.

Step 6

In the other order: $RS=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ .

Why

The same calculation with the factors swapped. Both results are reflections, but not about the same axis: the concrete proof of non-commutativity.

Step 7

$RS$ is the reflection about $y=x$ , and $SR$ the one about $y=-x$ : the two compositions are DIFFERENT.

Why

Here the two axes are perpendicular, and not by chance: the axes of $RS$ and $SR$ are separated by the angle of $R$ , namely $90^{\circ}$ . The rule confirms it: two reflections whose axes make an angle $\alpha$ compose into a rotation through $2\alpha$ , and indeed $(RS)(SR)=RS^{2}R=R^{2}$ , the rotation through $180^{\circ}$ .

Step 8

Check on $(1, 0)$ : $R$ sends it to $(0, 1)$ , then $S$ to $(0, -1)$ . And $SR\begin{pmatrix} 1 \\ 0 \end{pmatrix}=\begin{pmatrix} 0 \\ -1 \end{pmatrix}$ .

Why

Following a vector by hand is the only check that tests the ORDER itself. Had the product been written the wrong way round, the result would be $(0, 1)$ and the error would jump out.

The conclusion, written out

“First $R$ then $S$ ” has matrix $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$ , the reflection about $y=-x$ ; “first $S$ then $R$ ” has matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ , the reflection about $y=x$ . The two compositions do not coincide.

The classic mistake on this problem: Writing $RS$ for “first $R$ , then $S$ ”. The result is a genuine reflection, about $y=x$ instead of $y=-x$ : nothing in the calculation flags the error; only the check on $(1, 0)$ reveals it.

Learn by heart

• The COLUMNS of $A$ are $T(1, 0)$ and $T(0, 1)$ , in that order.
• $T$ linear implies $T(\vec{0})=\vec{0}$ : a translation is never linear.
• $R_{\theta}=\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ , with the minus sign AT THE TOP right, determinant $1$ .
• Reflection about the axis at angle $\theta$ : DOUBLE angle in the matrix, determinant $-1$ .
• Projection: determinant $0$ . Shear: determinant $1$ without being a rotation.
• $T_{2}\circ T_{1}$ has matrix $A_{2}A_{1}$ : the FIRST one applied is on the RIGHT.
• Area of the image $=|\det(A)|\times$ original area; negative determinant, orientation reversed.
• $R_{\theta}^{n}=R_{n\theta}$ : composing rotations means adding the angles.
• Axis of a reflection: solve $A\vec{v}=\vec{v}$ .
• Two reflections whose axes make an angle $\alpha$ compose into a rotation through $2\alpha$ .

Frequently asked questions

How do you find the matrix of a transformation of the plane?

Compute the image of the vector one, zero and the image of the vector zero, one, then put these two images in columns. The first image becomes the first column, the second image the second column. This rule holds only for the vectors of the standard basis, not for any two vectors.

In which order do you multiply the matrices of a composition?

The transformation applied first goes on the right of the product. If you rotate and then reflect, the matrix is the reflection matrix multiplied by the rotation matrix. The reason is that the right-hand factor is the one that meets the vector first in the calculation.

Why does the reflection matrix use the double angle?

Because a point at a given angle is sent to the other side of the axis, at the same angular distance. Its new angle is therefore twice the angle of the axis minus its starting angle. It is the gap that doubles, whereas a rotation simply adds its angle.

What does a zero determinant mean for a transformation?

That the transformation flattens the plane onto a line or onto a point. It does exist, but it is not invertible: several different points have the same image, and the area of every transformed figure becomes zero. Projections are the typical example of this case.

How do you recognize a rotation from its matrix?

Its determinant is one, and its two columns have length one and are perpendicular to each other. The angle can then be read directly from the first column, whose coordinates are the cosine and the sine of that angle. A determinant of minus one would signal a reflection, not a rotation.

Is a translation a linear transformation?

No. A linear transformation always leaves the origin fixed, whereas a translation moves it. A translation is called affine: it is a linear transformation followed by the addition of a constant vector, and only the linear part can be represented by a matrix.

Practise it

Corrected exercises: Linear transformations of the plane, 201-SN4-RE

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.

16 corrected exercises
160 points
225 minutes

Do the exercises

Something missing from this sheet? A method, a trap, an exercise to practise it, a mistake you spotted: tell me, I read every request.

Sending goes through your free space: that is how I can answer you and tell you when the exercise is added. An email is enough, no password. Create your space. Rather not create an account? Write to me directly.

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© Ahmed Squalli Houssaini. Revision sheet published at www.letuteurscientifique.ca/en/fiches/cegep-sn4-linear-transformations. Free for personal and classroom use; republishing it elsewhere requires written permission (legal notice).

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Revision sheet: linear transformations of the plane (201-SN4-RE)

The essentials

The columns are the images of the standard basis

The catalogue, with the determinants that identify

Composing, and reading the area factor

The mistakes that cost marks

1. Composing in reading order instead of matrix order

2. Putting the minus sign on the wrong side of the rotation matrix

3. Using the single angle for a reflection

4. Mixing up degrees and radians

5. Concluding that a transformation with zero determinant does not exist

6. Putting in columns the images of vectors that are not the basis vectors

7. Taking a translation for a linear transformation

8. Raising a matrix to a power entry by entry

Which method to choose

Which transformation, from the matrix alone

In which order to write the product

How the answer is expected to be written

Finding the matrix of a transformation described geometrically

Identifying a transformation from its matrix

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The typical problem, taken apart

A rotation and a reflection, composed in both orders

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Frequently asked questions

How do you find the matrix of a transformation of the plane?

In which order do you multiply the matrices of a composition?

Why does the reflection matrix use the double angle?

What does a zero determinant mean for a transformation?

How do you recognize a rotation from its matrix?

Is a translation a linear transformation?

Corrected exercises: Linear transformations of the plane, 201-SN4-RE

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