AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/4/4-12)
Unit 4 · Topic 4.12
4.12 Linear Transformations and Matrices
A linear transformation takes in a vector and gives back a vector, and each output component is built only by adding constant multiples of the input components. Every linear transformation of the plane can be written as multiplication by a 2 × 2 matrix, and it always sends the zero vector to itself.
Key terms
- linear transformation
- transformation matrix
- input vector
- output vector
What makes a transformation linear
A linear transformation L takes a vector ⟨x, y⟩ and returns a vector whose components each have the form (constant)·x + (constant)·y. For example, L(⟨x, y⟩) = ⟨2x − y, x + 3y⟩ is linear.
No constant terms, squares, products like xy, or other functions are allowed. ⟨x + 1, y⟩ and ⟨x², y⟩ are not linear.
Because there are no constant terms, a linear transformation always maps the zero vector to the zero vector: L(⟨0, 0⟩) = ⟨0, 0⟩. If a transformation moves the origin, it isn't linear.
Matrices and linear transformations
Write a vector as a 2 × 1 column. Then L(⟨x, y⟩) = ⟨ax + by, cx + dy⟩ is exactly the matrix product [a b; c d][x; y].
So every linear transformation from the plane to the plane has exactly one 2 × 2 matrix A with L(v) = Av. Going the other way, every 2 × 2 matrix A defines a linear transformation L(v) = Av.
To find the matrix, read the coefficients of x and y from each output component, row by row.
Transforming many vectors at once
A set of n vectors can be written as a 2 × n matrix, with each vector as a column. Multiplying the 2 × 2 transformation matrix by this 2 × n matrix gives a 2 × n matrix whose columns are the n output vectors.
This is how you transform a whole shape: put its corner points in as columns, multiply, and read off the new corners.
Common linear transformations
A linear transformation keeps the origin fixed and sends straight lines to straight lines (or to a single point). It can stretch, shrink, reflect, rotate or shear the plane, but it can never slide the plane over.
| Matrix | What it does |
|---|---|
| [1 0; 0 −1] | Reflects over the x-axis |
| [−1 0; 0 1] | Reflects over the y-axis |
| [0 1; 1 0] | Reflects over the line y = x |
| [k 0; 0 k] | Dilates by a factor of k |
| [k 0; 0 1] | Stretches horizontally by a factor of k |
| [1 k; 0 1] | Shears horizontally: slides each point sideways by k times its height |
Worked examples
Try each one yourself first, then open the solution.
- Example 1
From formula to matrix
Write the matrix for L(⟨x, y⟩) = ⟨2x − y, x + 3y⟩ and find L(⟨4, −1⟩).
Show the solutionHide the solution
- Step 1: Row 1 holds the coefficients of the first output, 2x − y: [2 −1]. Row 2 holds those of x + 3y: [1 3].
- Step 2: A = [2 −1; 1 3].
- Step 3: A[4; −1] = [2(4) + (−1)(−1); 1(4) + 3(−1)] = [9; 1].
Answer: A = [2 −1; 1 3]; L(⟨4, −1⟩) = ⟨9, 1⟩.
- Example 2
Trap: a shift is not linear
Is T(⟨x, y⟩) = ⟨x + 1, y⟩ a linear transformation?
Show the solutionHide the solution
- Step 1: It looks simple, but check the zero vector: T(⟨0, 0⟩) = ⟨1, 0⟩.
- Step 2: A linear transformation must send ⟨0, 0⟩ to ⟨0, 0⟩. This one doesn't, because of the constant +1.
- Step 3: So there's no 2 × 2 matrix A with T(v) = Av.
Answer: No. It moves the origin, so it isn't linear.
- Example 3
Transforming a triangle
Apply A = [1 1; 0 2] to the triangle with corners (0, 0), (2, 0) and (1, 3).
Show the solutionHide the solution
- Step 1: Put the corners in as columns: [0 2 1; 0 0 3].
- Step 2: Multiply: row 1 of A is [1 1], giving 0 + 0 = 0, 2 + 0 = 2, 1 + 3 = 4. Row 2 is [0 2], giving 0, 0, 6.
- Step 3: The product is [0 2 4; 0 0 6].
Answer: The new corners are (0, 0), (2, 0) and (4, 6).
Common mistakes
- Calling a transformation with a constant term linear.
- Putting the coefficients of one output into a column instead of a row.
- Multiplying in the wrong order: the transformation matrix goes on the left, A times v.
On the exam
- Unit 4 is not on the AP Precalculus Exam, so you'll see this topic on class tests rather than in May.
- Expect to write the matrix for a linear transformation, apply it to vectors or shapes, and decide whether a transformation is linear.
Connected topics
Videos
Check yourself
3 questions on 4.12 Linear Transformations and Matrices. Pick an answer to see if you got it, and why.
The linear transformation L is given by L(⟨x, y⟩) = ⟨2x − y, x + 3y⟩. What is L(⟨1, 2⟩)?
Matrices are written row by row, with rows separated by semicolons. The linear transformation L is given by L(⟨x, y⟩) = ⟨3x − y, 2y⟩. Which matrix represents L, so that L(v) is the matrix times the column vector v?
Which of the following transformations of the plane cannot be a linear transformation?
0 of 3 answered