Unit 4 · Topic 4.12 · about 30 minutes · 7 parts · 13 scored checks, 3 of them multiple choice

AP Precalculus 4.12: Linear Transformations and Matrices

Built by Tanner Crow from College Board's course framework · Every answer key independently checked

Not assessed on the AP Exam. Unit 4 is part of the course, but nothing in it is on the exam.

A dashed triangle has corners (1, 1), (3, 1) and (2, 3). The matrix A with rows 2, -1 and 1, 1 times the 2 by 3 matrix V with those corners as its columns gives AV, whose columns (1, 2), (5, 4) and (1, 5) are the corners of the shaded image triangle, in the same order. A dotted line joins each corner to its image.

Part 1 Think first Which filter is a matrix?

Guess before you learn. A wrong guess still makes the answer stick.

Predict first

A drawing app has two filters that move every point of a picture. Filter P sends the point $(x, y)$ to $(x + 3, 2 y)$ . Filter Q sends $(x, y)$ to $(3 x, x + 2 y)$ . Which filter could the app run by writing each point as a column and multiplying it by one $2 \times 2$ matrix?

In short. A linear transformation $L$ from $ℝ^{2}$ to $ℝ^{2}$ is a function whose output components are sums of constant multiples of the input components, and it is always $L (\vec{v}) = A \vec{v}$ for exactly one $2 \times 2$ matrix $A$ .

$L (⟨ x, y ⟩) = ⟨ a x + b y, c x + d y ⟩$ has the matrix $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ : row 1 holds the multiples in the first output component, and $x$ 's multiple sits in column 1.
Every linear transformation sends the zero vector to the zero vector, so a rule that moves $⟨ 0, 0 ⟩$ is not one.
Written as the columns of a $2 \times n$ matrix $V$ , $n$ input vectors go through $L$ all at once: $A V$ is a $2 \times n$ matrix whose columns are the $n$ output vectors.

Part 2 Learn it Functions that take vectors

2a. Outputs made from the inputs

A function can take a vector in and give a vector out. A linear transformation from $ℝ^{2}$ to $ℝ^{2}$ does it in one particular way: each component of the output vector is a sum of constant multiples of the input vector's components.

Here is one:

L (⟨ x, y ⟩) = ⟨ 2 x + y, x - 3 y ⟩

The first output component is 2 times $x$ plus 1 times $y$ . The second is 1 times $x$ plus $- 3$ times $y$ . For the input $⟨ 4, 1 ⟩$ , the output is $⟨ 8 + 1, 4 - 3 ⟩ = ⟨ 9, 1 ⟩$ .

A constant can be 0 or negative, so $⟨ 5 y, - x ⟩$ and $⟨ x, 0 ⟩$ qualify too. What breaks the pattern is a term that is not a constant times $x$ or $y$ : a constant added on, as in $x + 2$ , or $x^{2}$ , or $x y$ .

Put in the zero vector, $⟨ 0, 0 ⟩$ . Every term is a constant times 0, so every output component is 0. A linear transformation always maps the zero vector to the zero vector, so a rule that moves $⟨ 0, 0 ⟩$ is not linear. Passing the test proves nothing, though. $⟨ x^{2}, y ⟩$ sends $⟨ 0, 0 ⟩$ to itself and is still not linear.

Quick check · Sort it

Each card is a function from $ℝ^{2}$ to $ℝ^{2}$ . Tap a card, then tap whether it is a linear transformation.

Linear transformation

Not a linear transformation

2b. The matrix behind the rule

A single vector in $ℝ^{2}$ can be written as a $2 \times 1$ matrix: $⟨ x, y ⟩$ becomes $[\begin{matrix} x \\ y \end{matrix}]$ . Now multiply a $2 \times 2$ matrix by it, row times column as in 4.10:

[\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} a x + b y \\ c x + d y \end{matrix}]

That is a linear transformation. So any $2 \times 2$ matrix $A$ gives one, $L (\vec{v}) = A \vec{v}$ . It works the other way too: every linear transformation $L$ from $ℝ^{2}$ to $ℝ^{2}$ has exactly one $2 \times 2$ matrix $A$ with $L (\vec{v}) = A \vec{v}$ .

To find it, line up the rule with $a x + b y$ and $c x + d y$ . For $L (⟨ x, y ⟩) = ⟨ 2 x + y, x - 3 y ⟩$ , row 1 is $2, 1$ and row 2 is $1, - 3$ :

A = [\begin{matrix} 2 & 1 \\ 1 & - 3 \end{matrix}], A [\begin{matrix} 4 \\ 1 \end{matrix}] = [\begin{matrix} 9 \\ 1 \end{matrix}]

That matches $L (⟨ 4, 1 ⟩) = ⟨ 9, 1 ⟩$ from before.

Read each component by its letters, not by its order on the page. If a rule says $y + 4 x$ , the 4 goes in column 1, because it multiplies $x$ .

Quick check · Multiple choice

A linear transformation is $L (⟨ x, y ⟩) = ⟨ 3 x - 2 y, y + 5 x ⟩$ . Which matrix $A$ gives $L (\vec{v}) = A \vec{v}$ , with $\vec{v}$ written as a $2 \times 1$ matrix?

2c. Many vectors in one product

Often you want the outputs for several vectors at once, such as every corner of a shape. Write the $n$ vectors as the columns of one $2 \times n$ matrix. For $⟨ 1, 2 ⟩$ , $⟨ 3, 0 ⟩$ and $⟨ - 1, 4 ⟩$ :

V = [\begin{matrix} 1 & 3 & - 1 \\ 2 & 0 & 4 \end{matrix}]

Multiply the $2 \times 2$ matrix $A$ on the left. The sizes read $(2 \times 2) (2 \times 3)$ , so $A V$ is $2 \times 3$ , and each column of $A V$ is $A$ times the matching column of $V$ . With $A = [\begin{matrix} 2 & 1 \\ 1 & - 3 \end{matrix}]$ from before:

A V = [\begin{matrix} 4 & 6 & 2 \\ - 5 & 3 & - 13 \end{matrix}]

Column 3 is the output for $⟨ - 1, 4 ⟩$ : $2 (- 1) + 1 (4) = 2$ and $1 (- 1) + (- 3) (4) = - 13$ , so $L (⟨ - 1, 4 ⟩) = ⟨ 2, - 13 ⟩$ .

The vectors go in as columns, and the outputs come out as columns in the same order. Write the vectors as rows instead and the product either does not fit or pairs the wrong numbers.

Quick check · Calculate

Let $A = [\begin{matrix} 2 & - 3 \\ 1 & 4 \end{matrix}]$ , and let $V$ be the $2 \times 3$ matrix whose columns are the vectors $⟨ 3, 1 ⟩$ , $⟨ - 2, 5 ⟩$ and $⟨ 1, 2 ⟩$ , in that order. What is the component in row 2, column 3 of $A V$ ?

Your answer

Part 3 Watch it done Move a triangle with one product

Watch each step, then do the last one yourself.

A design program applies $L (⟨ x, y ⟩) = ⟨ x + 2 y, 3 x - y ⟩$ to a triangle with corners $(2, 1)$ , $(1, 3)$ and $(3, - 1)$ . Each corner is the vector from the origin to it. Find all three image corners with one matrix product.

Step 1 · Write the matrix
Row 1 holds the multiples in $x + 2 y$ , and row 2 those in $3 x - y$ : $A = [\begin{matrix} 1 & 2 \\ 3 & - 1 \end{matrix}]$ .
Step 2 · Put the corners in columns
$C = [\begin{matrix} 2 & 1 & 3 \\ 1 & 3 & - 1 \end{matrix}]$ , a $2 \times 3$ matrix. The product $A C$ will be $2 \times 3$ , one column per corner.
Why this step?
Why do the corners go in as columns, with A on the left?
$L (\vec{v}) = A \vec{v}$ needs $\vec{v}$ as a $2 \times 1$ column. Three columns side by side make a $2 \times 3$ matrix that fits after a $2 \times 2$ one, and each column of the product is $A$ times one corner.
Step 3 · Multiply column by column
Column 1: $1 (2) + 2 (1) = 4$ and $3 (2) + (- 1) (1) = 5$ . Column 2: $1 (1) + 2 (3) = 7$ and $3 (1) + (- 1) (3) = 0$ . Column 3 starts with $1 (3) + 2 (- 1) = 1$ .
Your step · Calculate

Finish it. With $A = [\begin{matrix} 1 & 2 \\ 3 & - 1 \end{matrix}]$ and the triangle's corners in the columns of $C = [\begin{matrix} 2 & 1 & 3 \\ 1 & 3 & - 1 \end{matrix}]$ , what is the component in row 2, column 3 of $A C$ , the second component of the image of the corner $(3, - 1)$ ?

Your answer

So $A C = [\begin{matrix} 4 & 7 & 1 \\ 5 & 0 & 10 \end{matrix}]$ , and the image triangle has corners $(4, 5)$ , $(7, 0)$ and $(1, 10)$ , in the same order as the corners that went in. One product moved the whole triangle.

Part 4 Try it Slant a game sprite

Warm up in the lab. It holds the matrix $A = [\begin{matrix} 1 & 1 \\ 0 & 2 \end{matrix}]$ and draws the dashed unit square, the unit vectors $\vec{i}$ and $\vec{j}$ , and a dashed vector $\vec{v}$ . The image is not drawn. Type a few vectors into the boxes for $\vec{v}$ , such as $⟨ 2, 1 ⟩$ and $⟨ - 1, 1 ⟩$ . For each one, work out $A \vec{v}$ on paper and find that point on the grid. Then find where $A$ sends each corner of the unit square.

The questions that follow use a different matrix. Work them on paper.

Lab

Vectors and Matrices

Move $\vec{v}$ by typing its components or dragging its head. Say where $A \vec{v}$ lands before you move on to the next vector.

Open the full Vectors and Matrices lab

A game draws a flag-shaped sprite with corners $(0, 0)$ , $(0, 4)$ , $(3, 4)$ and $(3, 2)$ . To make it lean, the game applies $T (⟨ x, y ⟩) = ⟨ x - y, 2 x + y ⟩$ to every point.

Try it · Match

A game applies $T (⟨ x, y ⟩) = ⟨ x - y, 2 x + y ⟩$ to a sprite with corners $(0, 0)$ , $(0, 4)$ , $(3, 4)$ and $(3, 2)$ . Tap a corner, then tap its image under $T$ .

Try it · Decide and justify

A game designer wants to slide a sprite 4 units to the right, sending every point $(x, y)$ to $(x + 4, y)$ . Can the slide be written as $L (\vec{v}) = A \vec{v}$ for some $2 \times 2$ matrix $A$ ?

Which reason supports your answer?

Part 5 Explain it Put it in writing

Nothing you type leaves this page. Check your answer against the model and each point on the list.

Unit 4 is not on the AP Exam, so this is written practice for the course.

Written practice

Let $L (⟨ x, y ⟩) = ⟨ 4 x - y, 2 x + 3 y ⟩$ , a function from $ℝ^{2}$ to $ℝ^{2}$ .

(i) Explain why $L$ is a linear transformation, and determine the $2 \times 2$ matrix $A$ with $L (\vec{v}) = A \vec{v}$ .

(ii) Write the vectors $⟨ 1, 2 ⟩$ , $⟨ 0, - 1 ⟩$ and $⟨ 3, 1 ⟩$ as one matrix, and determine all three outputs of $L$ with one matrix product.

(iii) Let $K (⟨ x, y ⟩) = ⟨ 4 x - y + 1, 2 x + 3 y ⟩$ . Explain why $K$ is not a linear transformation.

Your answer stays on this page. It is never sent or saved.

Model answer

(i) Each output component is a sum of constant multiples of $x$ and $y$ : $4 x + (- 1) y$ and $2 x + 3 y$ . So $L$ is linear, and its matrix holds those multiples row by row: $A = [\begin{matrix} 4 & - 1 \\ 2 & 3 \end{matrix}]$ . (ii) With the vectors as columns, $V = [\begin{matrix} 1 & 0 & 3 \\ 2 & - 1 & 1 \end{matrix}]$ , a $2 \times 3$ matrix. Then $A V = [\begin{matrix} 2 & 1 & 11 \\ 8 & - 3 & 9 \end{matrix}]$ , so the outputs are $⟨ 2, 8 ⟩$ , $⟨ 1, - 3 ⟩$ and $⟨ 11, 9 ⟩$ . For example, column 3 is $4 (3) + (- 1) (1) = 11$ and $2 (3) + 3 (1) = 9$ . (iii) $K (⟨ 0, 0 ⟩) = ⟨ 1, 0 ⟩$ . A linear transformation sends the zero vector to the zero vector, so $K$ is not one. The $+ 1$ is not a constant multiple of $x$ or $y$ .

Score your own answer. This is a self-check, not a grade. Tick each point your answer makes.

Says each output component of $L$ is a sum of constant multiples of $x$ and $y$ , and gives $A = [\begin{matrix} 4 & - 1 \\ 2 & 3 \end{matrix}]$ .
Writes the three vectors as the columns of a $2 \times 3$ matrix and multiplies with $A$ on the left.
Gets the outputs $⟨ 2, 8 ⟩$ , $⟨ 1, - 3 ⟩$ and $⟨ 11, 9 ⟩$ , one per column.
Shows $K (⟨ 0, 0 ⟩) = ⟨ 1, 0 ⟩$ and explains that a linear transformation must send the zero vector to the zero vector.

Part 6 Check yourself A logo in a drawing app

The three case questions use the same logo and matrix. Answer them from the table.

A drawing app stores the four corners of a logo as the columns of a matrix $C$ , each corner as the vector from the origin to it. It transforms the logo with $L (\vec{v}) = M \vec{v}$ , and the image corners are the columns of $M C$ .

The corner matrix C and the transformation matrix M
Name	Matrix
$C$	$[\begin{matrix} 0 & 3 & 3 & 1 \\ 0 & 0 & 2 & 2 \end{matrix}]$
$M$	$[\begin{matrix} 1 & - 2 \\ 2 & 1 \end{matrix}]$

Case question 1 · Multiple choice

A drawing app transforms a logo with corners $(0, 0)$ , $(3, 0)$ , $(3, 2)$ and $(1, 2)$ by $L (\vec{v}) = M \vec{v}$ , where $M = [\begin{matrix} 1 & - 2 \\ 2 & 1 \end{matrix}]$ . Which corner is its own image?

Case question 2 · Multiple choice

A logo's corners are the columns of $C = [\begin{matrix} 0 & 3 & 3 & 1 \\ 0 & 0 & 2 & 2 \end{matrix}]$ , in order. The transformation $L (\vec{v}) = M \vec{v}$ , with $M = [\begin{matrix} 1 & - 2 \\ 2 & 1 \end{matrix}]$ , sends them to the columns of $M C$ . Read as a point, which image corner is column 3 of $M C$ ?

Case question 3 · Spot the error

A student finds the image of the logo's corner $(1, 2)$ under $L (\vec{v}) = M \vec{v}$ , with $M = [\begin{matrix} 1 & - 2 \\ 2 & 1 \end{matrix}]$ . Tap the first line that goes wrong.

The rest stand on their own and do not use the case.

Check yourself · Fill the statement

A linear transformation is $L (\vec{v}) = A \vec{v}$ for a $2 \times 2$ matrix $A$ . You want its outputs for five vectors from $ℝ^{2}$ at once. Tap a blank, then tap an answer.

Written as the columns of one matrix $V$ , the five vectors make a matrix. The product holds the five output vectors, and its column 4 is .

Check yourself · Select all

Let $A$ be any $2 \times 2$ matrix, and let $L (\vec{v}) = A \vec{v}$ for vectors $\vec{v}$ in $ℝ^{2}$ . Which two statements must be true? Pick two.

Select 2 answers, then check.

Part 7 Keep it Bring back earlier ideas

Pulling an older idea back up is what makes it last.

From 4.11 The Inverse and Determinant of a Matrix · Calculate

What is the area, in square units, of the parallelogram spanned by the vectors $⟨ 1, 4 ⟩$ and $⟨ 3, 1 ⟩$ ?

Your answer

From 2.2 Change in Linear and Exponential Functions · Calculate

A linear function has slope 3 and $f (4) = 17$ . What is $f (10)$ ?

Your answer

In three lines

A linear transformation makes each output component a sum of constant multiples of the input components, and sends $⟨ 0, 0 ⟩$ to $⟨ 0, 0 ⟩$ .
$L (⟨ x, y ⟩) = ⟨ a x + b y, c x + d y ⟩$ is $L (\vec{v}) = A \vec{v}$ with $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ , the one matrix that does it.
Put $n$ input vectors in the columns of a $2 \times n$ matrix $V$ ; the columns of $A V$ are their outputs, in the same order.

Vocabulary

Linear transformation: A function that maps an input vector to an output vector so that each component of the output vector is the sum of constant multiples of the input vector components.
Input vector and output vector: For a linear transformation $L$ , the vector $\vec{v}$ that goes in and the vector $L (\vec{v})$ that comes out.
Zero vector: The vector $⟨ 0, 0 ⟩$ , which every linear transformation maps to itself.
Transformation matrix: The unique $2 \times 2$ matrix $A$ with $L (\vec{v}) = A \vec{v}$ for every $\vec{v}$ in $ℝ^{2}$ , for a linear transformation $L$ from $ℝ^{2}$ to $ℝ^{2}$ .
2 × n matrix of vectors: $n$ vectors in $ℝ^{2}$ written as the $n$ columns of one matrix, each vector a $2 \times 1$ column.

Course alignment, for teachers

AP Precalculus topic 4.12, Unit 4: Functions Involving Parameters, Vectors, and Matrices.

Skill 1.B: Express functions, equations, or expressions in analytically equivalent forms that are useful in a given mathematical or applied context.

This lesson maps all 5 essential knowledge statements of the topic to the part that teaches each one: 5 are checked by a scored question. The map is in the lesson file.

Part 1 Think first Which filter is a matrix?

Part 2 Learn it Functions that take vectors

2a. Outputs made from the inputs

2b. The matrix behind the rule

2c. Many vectors in one product

Part 3 Watch it done Move a triangle with one product

Part 4 Try it Slant a game sprite

Vectors and Matrices

Part 5 Explain it Put it in writing

Part 6 Check yourself A logo in a drawing app

Part 7 Keep it Bring back earlier ideas

Vocabulary

Labs that use this topic