4.6: Linear Transformations | AP Precalculus

How it works

A matrix maps column vectors to column vectors. Its columns are the images of the standard basis vectors. Linear transformations preserve the origin; a translation by a nonzero vector is not linear.

T(\mathbf{x})=A\mathbf{x},\quad T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})

Worked examples

Example 1

Apply $\displaystyle \begin{pmatrix}2&0\\0&1\end{pmatrix}$ to $\displaystyle (3,4)$ .

The image is $\displaystyle (6,4)$ , a horizontal stretch by factor $\displaystyle 2$ .

Example 2

Give the matrix of a $\displaystyle 90^\circ$ counterclockwise rotation about the origin.

It is $\displaystyle \begin{pmatrix}0&-1\\1&0\end{pmatrix}$ because $\displaystyle (1,0)$ maps to $\displaystyle (0,1)$ and $\displaystyle (0,1)$ maps to $\displaystyle (-1,0)$ .

Problems

Problem 4.6.1

Apply the $\displaystyle 90^\circ$ counterclockwise rotation matrix to the vector $\displaystyle (3,1)$ .

Vector from 0,0 to 3,1; the solution shows its image at negative 1,3. — A rotation about the origin maps (x,y) to (-y,x).

Show worked solution

Matrix multiplication gives $\displaystyle (-1,3)$ . Its length remains $\displaystyle \sqrt{10}$ .

Vector from 0,0 to 3,1; the solution shows its image at negative 1,3. The solution adds labeled relationships. — The matrix maps (3,1) to (-1,3) and preserves length.

Problem 4.6.2

Give the matrix of reflection across the x-axis and apply it to $\displaystyle (2,5)$ .

Show worked solution

The matrix is $\displaystyle \begin{pmatrix}1&0\\0&-1\end{pmatrix}$ and the image is $\displaystyle (2,-5)$ .

Problem 4.6.3

Explain why $\displaystyle T(x,y)=(x+2,y)$ is not linear.

Show worked solution

It sends $\displaystyle (0,0)$ to $\displaystyle (2,0)$ . A linear transformation must send the zero vector to itself.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 4.6.1

Problem 4.6.2

Problem 4.6.3

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