Topic 4 of 8

Linear transformations

Step 19 · Linear algebra topics

See a matrix as a function that moves vectors, recognize rotations, reflections, stretches and projections, and find what a transformation sends to zero and what it can reach.

Matrices as transformations

A transformation T is linear when T(u + v) = T(u) + T(v) and T(cu) = cT(u). Every linear transformation of ℝⁿ is multiplication by a matrix, and the matrix's columns are where the basis vectors land.

Example: Reading the columns

Where does $\begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$ send ⟨1, 0⟩ and ⟨0, 1⟩?

Multiplying by ⟨1, 0⟩ picks out the first column.
Multiplying by ⟨0, 1⟩ picks out the second.

AnswerTo ⟨2, 0⟩ and ⟨1, 3⟩

Example: Building a matrix

T stretches the x direction by 3 and leaves y alone. Find its matrix.

T⟨1, 0⟩ = ⟨3, 0⟩ and T⟨0, 1⟩ = ⟨0, 1⟩.
Those are the columns.

Answer $\begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}$

Geometric transformations

Transformation	Matrix
Turn by θ	$\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$
Reflect in the x-axis	$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
Stretch by a and b	$\begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}$
Project onto the x-axis	$\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$
Shear	$\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$

The shear with columns ⟨1, 0⟩ and ⟨1, 1⟩ slides the unit square into a parallelogram with the same base and height, so the same area.

Example: A rotation

Turn ⟨1, 0⟩ by 90° counterclockwise using a matrix.

cos 90° = 0 and sin 90° = 1, so the matrix is $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ .
Its first column is where ⟨1, 0⟩ lands.

Answer⟨0, 1⟩

Kernel and range

The kernel of T is everything it sends to 0; the range is everything it can reach. For a matrix these are its null space and column space. T is one-to-one exactly when its kernel is only 0.

Example: A projection

Find the kernel and range of $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ .

It sends ⟨x, y⟩ to ⟨x, 0⟩.
⟨x, y⟩ goes to 0 exactly when x = 0.

AnswerKernel: the y-axis. Range: the x-axis.

Example: Counting dimensions

T is given by a 3 × 3 matrix with 2 pivots. What are the dimensions of its kernel and range?

The range has dimension 2, one per pivot.
The kernel has dimension 3 − 2 = 1.

AnswerKernel 1, range 2: T squashes space onto a plane

Try it yourself

Work each one on paper first, then open the answer to check.

Where does $\begin{pmatrix} 1 & 4 \\ 2 & 5 \end{pmatrix}$ send ⟨0, 1⟩?
Show answer
⟨4, 5⟩
Find the matrix that reflects the plane in the y-axis.
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$\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$
Find the matrix that turns the plane by 180°.
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$\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$
Is T⟨x, y⟩ = ⟨x + 1, y⟩ linear?
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No: it moves 0 to ⟨1, 0⟩
What is the kernel of $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ ?
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The line y = −x
A 2 × 3 matrix has 2 pivots. What is the dimension of its kernel?
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1