Unit 4 · Topic 4.12

Linear Transformations and Matrices

A linear transformation sends ⟨x, y⟩ to ⟨ax + by, cx + dy⟩ , which is the matrix [a, b; c, d] times the input.

Practice 2: Multiple Representations

Beyond the exam. The AP® Precalculus exam does not test this unit. Many classes teach it, and it prepares you for calculus.

Part 1 · Hook

Why this matters

Photo apps rotate, flip, stretch and slant pictures. Each of those moves sends every point of the image to a new point by the same rule, keeps the center fixed and keeps straight lines straight. Every such move is a matrix, and the matrix can be read straight off a picture of where the square goes.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. What is [0−110][31]\left[ \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right]\left[ \begin{matrix} 3 \\ 1 \end{matrix} \right]?

  1. [−13]\left[ \begin{matrix} -1 \\ 3 \end{matrix} \right]
  2. [1−3]\left[ \begin{matrix} 1 \\ -3 \end{matrix} \right]
  3. [3−1]\left[ \begin{matrix} 3 \\ -1 \end{matrix} \right]
  4. [03]\left[ \begin{matrix} 0 \\ 3 \end{matrix} \right]
Show the answer

Row 1: 0(3)+(−1)(1)=−10(3) + (-1)(1) = -1. Row 2: 1(3)+0(1)=31(3) + 0(1) = 3.

  • Correct: [−13]\left[ \begin{matrix} -1 \\ 3 \end{matrix} \right]:
  • [1−3]\left[ \begin{matrix} 1 \\ -3 \end{matrix} \right]:
  • [3−1]\left[ \begin{matrix} 3 \\ -1 \end{matrix} \right]:
  • [03]\left[ \begin{matrix} 0 \\ 3 \end{matrix} \right]:

2. What is det⁡[2−111]\det \left[ \begin{matrix} 2 & -1 \\ 1 & 1 \end{matrix} \right]?

  1. 33
  2. 11
  3. −3-3
  4. 22
Show the answer

2(1)−(−1)(1)2(1) - (-1)(1)=2+1= 2 + 1=3= 3.

  • Correct: 33:
  • 11:
  • −3-3:
  • 22:

Part 4 · See it

See it first

The unit square with corners (0, 0), (1, 0), (1, 1) and (0, 1), dashed in gray, and its image under the matrix with rows (1, 1) and (0, 2), shaded green: a parallelogram with corners (0, 0), (1, 0), (2, 2) and (1, 2). The image of <1, 0> is <1, 0> (blue arrow) and the image of <0, 1> is <1, 2> (vermillion arrow), the two columns of the matrix.
The matrix [1102]\left[ \begin{matrix} 1 & 1 \\ 0 & 2 \end{matrix} \right] sends ⟨1,0⟩\left\langle 1, 0 \right\rangle to its first column, ⟨1,0⟩\left\langle 1, 0 \right\rangle, and ⟨0,1⟩\left\langle 0, 1 \right\rangle to its second column, ⟨1,2⟩\left\langle 1, 2 \right\rangle. The unit square becomes a parallelogram of area 22, the determinant. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. L(⟨x,y⟩)L(\left\langle x, y \right\rangle)=⟨ax+by,cx+dy⟩= \left\langle ax + by, cx + dy \right\rangleLL is multiplication by the matrix [abcd]\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right], with the input written as a column
  2. Apply LL to ⟨1,0⟩\left\langle 1, 0 \right\rangle and to ⟨0,1⟩\left\langle 0, 1 \right\rangleyou get the first and second columns of the matrix
  3. Every point moves by the same matrixthe origin stays put, lines stay lines and the grid becomes a grid of parallelograms
  4. Take the determinant of the matrixits absolute value is the factor by which every area is multiplied; a negative sign means the picture is flipped

Part 6 · Key ideas

Key ideas

  • A linear transformation of the plane has the form L(⟨x,y⟩)L(\left\langle x, y \right\rangle)=⟨ax+by,cx+dy⟩= \left\langle ax + by, cx + dy \right\rangle: multiplication by [abcd]\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right].
  • The columns of the matrix are L(⟨1,0⟩)L(\left\langle 1, 0 \right\rangle) and L(⟨0,1⟩)L(\left\langle 0, 1 \right\rangle), so a picture of where the unit square goes gives the matrix.
  • Rotation by θ\theta counterclockwise: [cos⁡θ−sin⁡θsin⁡θcos⁡θ]\left[ \begin{matrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{matrix} \right]. Reflection across the xx-axis: [100−1]\left[ \begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix} \right]. Dilation by kk: [k00k]\left[ \begin{matrix} k & 0 \\ 0 & k \end{matrix} \right].
  • Areas are multiplied by |det⁡|\left| \det \right|; a negative determinant reverses orientation.

Part 7 · Misconception

A common mistake

The wrong idea: If L(⟨1,0⟩)=⟨2,1⟩L(\left\langle 1, 0 \right\rangle) = \left\langle 2, 1 \right\rangle and L(⟨0,1⟩)=⟨−1,1⟩L(\left\langle 0, 1 \right\rangle) = \left\langle -1, 1 \right\rangle, the matrix is [21−11]\left[ \begin{matrix} 2 & 1 \\ -1 & 1 \end{matrix} \right].

What actually happens: The images go in as columns, not rows: the matrix is [2−111]\left[ \begin{matrix} 2 & -1 \\ 1 & 1 \end{matrix} \right]. Check: it sends ⟨1,0⟩\left\langle 1, 0 \right\rangle to its first column, ⟨2,1⟩\left\langle 2, 1 \right\rangle.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

A square and its image

A linear transformation TT sends the unit square (dashed) to the shaded parallelogram. The arrows show T(⟨1,0⟩)T(\left\langle 1, 0 \right\rangle) and T(⟨0,1⟩)T(\left\langle 0, 1 \right\rangle).

The unit square, dashed in gray, and its image under a linear transformation, shaded green: a parallelogram with corners (0, 0), (2, 1), (1, 2) and (-1, 1). The image of <1, 0> is drawn as a blue arrow to (2, 1) and the image of <0, 1> as a vermillion arrow to (-1, 1).
LevlPrep original diagram.

1. What is the matrix of TT?

  1. [2−111]\left[ \begin{matrix} 2 & -1 \\ 1 & 1 \end{matrix} \right]
  2. [21−11]\left[ \begin{matrix} 2 & 1 \\ -1 & 1 \end{matrix} \right]
  3. [−1211]\left[ \begin{matrix} -1 & 2 \\ 1 & 1 \end{matrix} \right]
  4. [2−11−1]\left[ \begin{matrix} 2 & -1 \\ 1 & -1 \end{matrix} \right]
Show the answer

The columns are the images: T(⟨1,0⟩)=⟨2,1⟩T(\left\langle 1, 0 \right\rangle) = \left\langle 2, 1 \right\rangle and T(⟨0,1⟩)=⟨−1,1⟩T(\left\langle 0, 1 \right\rangle) = \left\langle -1, 1 \right\rangle.

  • Correct: [2−111]\left[ \begin{matrix} 2 & -1 \\ 1 & 1 \end{matrix} \right]: Images as columns.
  • [21−11]\left[ \begin{matrix} 2 & 1 \\ -1 & 1 \end{matrix} \right]: The images were written as rows; they go in as columns.
  • [−1211]\left[ \begin{matrix} -1 & 2 \\ 1 & 1 \end{matrix} \right]: The columns are in the wrong order: the image of the first unit vector comes first.
  • [2−11−1]\left[ \begin{matrix} 2 & -1 \\ 1 & -1 \end{matrix} \right]: T(⟨0,1⟩)=⟨−1,1⟩T(\left\langle 0, 1 \right\rangle) = \left\langle -1, 1 \right\rangle, so the bottom right entry is 11.

2. What is T(⟨3,2⟩)T(\left\langle 3, 2 \right\rangle)?

  1. ⟨4,5⟩\left\langle 4, 5 \right\rangle
  2. ⟨8,−1⟩\left\langle 8, -1 \right\rangle
  3. ⟨5,5⟩\left\langle 5, 5 \right\rangle
  4. ⟨6,2⟩\left\langle 6, 2 \right\rangle
Show the answer

[2−111][32]\left[ \begin{matrix} 2 & -1 \\ 1 & 1 \end{matrix} \right]\left[ \begin{matrix} 3 \\ 2 \end{matrix} \right]=[45]= \left[ \begin{matrix} 4 \\ 5 \end{matrix} \right]: 33 copies of ⟨2,1⟩\left\langle 2, 1 \right\rangle plus 22 copies of ⟨−1,1⟩\left\langle -1, 1 \right\rangle.

  • Correct: ⟨4,5⟩\left\langle 4, 5 \right\rangle: 6−2=46 - 2 = 4 and 3+2=53 + 2 = 5.
  • ⟨8,−1⟩\left\langle 8, -1 \right\rangle: This uses the images as rows instead of columns.
  • ⟨5,5⟩\left\langle 5, 5 \right\rangle: 2(3)+(−1)(2)=42(3) + (-1)(2) = 4, not 55.
  • ⟨6,2⟩\left\langle 6, 2 \right\rangle: This multiplies matching entries; use row times column.

3. What is the area of the shaded parallelogram?

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

The unit square (area 11) is scaled by det⁡[2−111]\det \left[ \begin{matrix} 2 & -1 \\ 1 & 1 \end{matrix} \right]=2+1= 2 + 1=3= 3.

  • Answer: 3

4. A triangle of area 55 is transformed by TT. What is the area of its image?

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

Every area is multiplied by |det⁡|=3\left| \det \right| = 3: 5⋅3=155 \cdot 3 = 15.

  • Answer: 15

5. Which matrix rotates the plane by π2\frac{\pi}{2} counterclockwise?

  1. [0−110]\left[ \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right]
  2. [01−10]\left[ \begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} \right]
  3. [100−1]\left[ \begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix} \right]
  4. [−100−1]\left[ \begin{matrix} -1 & 0 \\ 0 & -1 \end{matrix} \right]
Show the answer

A quarter turn counterclockwise sends ⟨1,0⟩\left\langle 1, 0 \right\rangle to ⟨0,1⟩\left\langle 0, 1 \right\rangle and ⟨0,1⟩\left\langle 0, 1 \right\rangle to ⟨−1,0⟩\left\langle -1, 0 \right\rangle: those are the columns.

  • Correct: [0−110]\left[ \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right]: Columns: images of the two unit vectors.
  • [01−10]\left[ \begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} \right]: This sends ⟨1,0⟩\left\langle 1, 0 \right\rangle to ⟨0,−1⟩\left\langle 0, -1 \right\rangle: a clockwise quarter turn.
  • [100−1]\left[ \begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix} \right]: This is the reflection across the xx-axis.
  • [−100−1]\left[ \begin{matrix} -1 & 0 \\ 0 & -1 \end{matrix} \right]: This is a half turn, by π\pi.

6. Which matrix reflects the plane across the line y=xy = x?

  1. [0110]\left[ \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right]
  2. [100−1]\left[ \begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix} \right]
  3. [−1001]\left[ \begin{matrix} -1 & 0 \\ 0 & 1 \end{matrix} \right]
  4. [0−1−10]\left[ \begin{matrix} 0 & -1 \\ -1 & 0 \end{matrix} \right]
Show the answer

The reflection swaps the coordinates: ⟨1,0⟩→⟨0,1⟩\left\langle 1, 0 \right\rangle \to \left\langle 0, 1 \right\rangle and ⟨0,1⟩→⟨1,0⟩\left\langle 0, 1 \right\rangle \to \left\langle 1, 0 \right\rangle.

  • Correct: [0110]\left[ \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right]: Swap xx and yy.
  • [100−1]\left[ \begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix} \right]: This reflects across the xx-axis.
  • [−1001]\left[ \begin{matrix} -1 & 0 \\ 0 & 1 \end{matrix} \right]: This reflects across the yy-axis.
  • [0−1−10]\left[ \begin{matrix} 0 & -1 \\ -1 & 0 \end{matrix} \right]: This reflects across the line y=−xy = -x.

Part 9 · Summary

Summary

A linear transformation sends ⟨x,y⟩\left\langle x, y \right\rangle to ⟨ax+by,cx+dy⟩\left\langle ax + by, cx + dy \right\rangle, which is the matrix [abcd]\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] times the input. Its columns are the images of ⟨1,0⟩\left\langle 1, 0 \right\rangle and ⟨0,1⟩\left\langle 0, 1 \right\rangle. Rotations, reflections, dilations and shears are all linear, and every area is multiplied by the absolute value of the determinant.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic