Matrix Representation: Rotation by 45°

The recipe for a transformation matrix

Every linear transformation $T : \mathbb{R}^n \to \mathbb{R}^m$ can be represented by a matrix $A$ such that $T(\mathbf{x}) = A \mathbf{x}$. To construct $A$:

Apply $T$ to each standard basis vector; each result becomes a column of $A$. $A = \bigl[\; T(\mathbf{e}_1) \mid T(\mathbf{e}_2) \mid \cdots \mid T(\mathbf{e}_n) \;\bigr]$

The rotation matrix

A counter-clockwise rotation by angle $\theta$ in $\mathbb{R}^2$: $R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$

Step-by-step for $\theta = 45°$

Step 1 — Rotate $\mathbf{e}_1 = (1, 0)$.

Point $(1, 0)$ rotated by 45° lands at $(\cos 45°, \sin 45°) = (\tfrac{\sqrt{2}}{2}, \tfrac{\sqrt{2}}{2}) \approx (0.707, 0.707)$. So first column of $A$ is $(\cos 45°, \sin 45°)^T$.

Step 2 — Rotate $\mathbf{e}_2 = (0, 1)$.

Point $(0, 1)$ rotated by 45° CCW ends at $(-\sin 45°, \cos 45°) = (-\tfrac{\sqrt{2}}{2}, \tfrac{\sqrt{2}}{2}) \approx (-0.707, 0.707)$. So second column is $(-\sin 45°, \cos 45°)^T$.

Step 3 — Assemble. $R_{45°} = \begin{pmatrix} \cos 45° & -\sin 45° \\ \sin 45° & \cos 45° \end{pmatrix} = \begin{pmatrix} \tfrac{\sqrt{2}}{2} & -\tfrac{\sqrt{2}}{2} \\ \tfrac{\sqrt{2}}{2} & \tfrac{\sqrt{2}}{2} \end{pmatrix}$

Testing on a sample vector

Rotate $\mathbf{v} = (1, 0)$: $R_{45°} \mathbf{v} = \begin{pmatrix} \tfrac{\sqrt{2}}{2} \\ \tfrac{\sqrt{2}}{2} \end{pmatrix}$

Correct — $\mathbf{v}$ pointed along the x-axis, now points along $y = x$, 45° up. ✓

Rotate $\mathbf{w} = (3, 4)$ (length 5): $R_{45°} \mathbf{w} = \begin{pmatrix} \tfrac{\sqrt{2}}{2}(3) - \tfrac{\sqrt{2}}{2}(4) \\ \tfrac{\sqrt{2}}{2}(3) + \tfrac{\sqrt{2}}{2}(4) \end{pmatrix} = \begin{pmatrix} -\tfrac{\sqrt{2}}{2} \\ \tfrac{7\sqrt{2}}{2} \end{pmatrix}$

Length: $\sqrt{(-\tfrac{\sqrt{2}}{2})^2 + (\tfrac{7 \sqrt{2}}{2})^2} = \sqrt{\tfrac{1}{2} + \tfrac{49}{2}} = \sqrt{25} = 5$ ✓

Length preserved — a hallmark of rotations.

Why rotations preserve length

For any $\mathbf{v}$: $\|R_\theta \mathbf{v}\|^2 = \mathbf{v}^T R_\theta^T R_\theta \mathbf{v} = \mathbf{v}^T I \mathbf{v} = \|\mathbf{v}\|^2$

since $R_\theta^T R_\theta = I$. A matrix with this property is called orthogonal. Rotations and reflections are the only length-preserving linear maps.

Key properties of rotation matrices

• Determinant: $\det R_\theta = \cos^2 \theta + \sin^2 \theta = 1$.
• Inverse: $R_\theta^{-1} = R_{-\theta} = R_\theta^T$ — the inverse is the transpose.
• Composition: $R_\alpha R_\beta = R_{\alpha + \beta}$ — rotation matrices add their angles.
• Fixed point: only $\mathbf{0}$ is fixed (for $\theta \ne 2\pi k$).

Transformations by inspection

Some useful matrices built via "where do the basis vectors go":

• Scale x by 2: $(1, 0) \to (2, 0), (0, 1) \to (0, 1)$: $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$.
• Reflect across $y = x$: $(1, 0) \to (0, 1), (0, 1) \to (1, 0)$: $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$.
• Shear x by 1 unit per y: $(1, 0) \to (1, 0), (0, 1) \to (1, 1)$: $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$.

Common mistakes

• Putting the sine and cosine in the wrong spots. Double-check by rotating $\mathbf{e}_1$ and comparing.
• Assuming every matrix represents a rotation. Only orthogonal matrices with $\det = 1$ are rotations; $\det = -1$ orthogonals are reflections.
• Mixing up clockwise vs. counter-clockwise. Our formula is CCW; for CW, replace $\theta$ with $-\theta$ (which flips the signs on the $\sin$ entries).

Try it in the visualization

Drag the angle $\theta$ via a slider from 0° to 360°. A sample vector rotates live; the matrix entries update; the unit circle stays a unit circle, confirming length preservation.