Introduction

In this article, we explore two of the most visually intuitive linear transformations in the plane:

• Rotations — spinning objects around the origin
• Scaling — stretching or shrinking objects

You already know what a linear transformation is. Now we look at how specific matrices act on 2D space and how to interpret their geometric meaning.

What Are Rotations?

A rotation turns every point in the plane around the origin by some angle $\theta$.

The standard rotation matrix is: $$R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$ Key properties:

• Rotations preserve lengths.
• Rotations preserve angles.
• Rotations do not change the shape of objects — only their orientation.

Useful facts:

• A $90^\circ$ rotation uses $\theta = \frac{\pi}{2}$.
• A $180^\circ$ rotation uses $\theta = \pi$.
• A $0^\circ$ rotation is just the identity matrix.

What Is Scaling?

Scaling stretches or shrinks objects.

A simple scaling matrix: $$S = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}$$ Effects:

• If $a > 1$, the $x$‑direction is stretched.
• If $0 < a < 1$, the $x$‑direction is compressed.
• If $a < 0$, the $x$‑direction is flipped and scaled.

Same for $b$ in the $y$‑direction.

Special case: uniform scaling $$kI = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$$ This stretches or shrinks equally in all directions.

Combining Rotations and Scaling

Because matrices represent linear transformations, we can compose them:

• First scale, then rotate: $R_\theta S$
• First rotate, then scale: $S R_\theta$

These are usually not the same matrix — order matters.

Geometric Interpretation

To understand a matrix, look at what it does to:

• The basis vectors $e_1 = (1,0)$ and $e_2 = (0,1)$
• A simple shape (square, triangle, grid)
• Distances and angles

Rotations:

• Preserve shape
• Preserve area
• Move points along circular arcs

Scaling:

• Changes size
• May distort shapes
• Multiplies area by $ab$ for matrix $\begin{pmatrix}a&0\\0&b\end{pmatrix}$

Examples

1. Rotation by $90^\circ$

$$R_{\pi/2} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$ Effect: turns every point counterclockwise by $90^\circ$.

2. Scaling by 2 in $x$ and 1/2 in $y$

$$S = \begin{pmatrix} 2 & 0 \\ 0 & 1/2 \end{pmatrix}$$ Effect: stretches horizontally, compresses vertically.

3. Uniform scaling by 3

$$3I = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}$$ Effect: enlarges all shapes by a factor of 3.

Calculator

Exercises

(5–10 exercises, each with an import‑comment as required)

Exercises

1. Compute $R_{\pi/2}(1,2)$ using $R_{\pi/2} = \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}$.

   Solution

   $$R_{\pi/2}(1,2) = (0\cdot1 - 1\cdot2,\; 1\cdot1 + 0\cdot2) = (-2,1)$$
2. Describe the geometric effect of $S = \begin{pmatrix}3 & 0 \\ 0 & 1\end{pmatrix}$.

   Solution

   
   The matrix stretches space horizontally by a factor of 3 while leaving vertical lengths unchanged.
3. Determine whether the transformation $T(x,y) = (2x,\, -3y)$ is a scaling transformation.

   Solution

   
   Yes. It scales the $x$‑direction by 2 and the $y$‑direction by $-3$ (a flip + stretch).
4. Compute the result of applying a uniform scaling by $k=4$ to the vector $(1,-2)$.

   Solution

   $$4(1,-2) = (4,-8)$$
5. Compute $R_\theta(1,0)$ for $\theta = \frac{\pi}{3}$.

   Solution

   $$R_{\pi/3}(1,0) = (\cos(\pi/3),\; \sin(\pi/3)) = \left(\frac12,\; \frac{\sqrt{3}}{2}\right)$$
6. True or false: A rotation can change the length of a vector.

   Solution

   
   False. Rotations preserve length.
7. Compute the composition $R_{\pi/2} S$ where $S = \begin{pmatrix}2 & 0 \\ 0 & 2\end{pmatrix}$.

   Solution

   $$S = \begin{pmatrix}2 & 0 \\ 0 & 2\end{pmatrix}$$ So $$R_{\pi/2} S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix}$$