Week 2: Linear Transformations and Matrices

What Is a Linear Transformation#

A linear transformation (or linear map) is a function $T$ from one vector space to another that respects the two vector-space operations: addition and scalar multiplication. Formally, $T$ is linear when both of the following hold for all vectors $u, v$ and every scalar $c$:

Together these say: “transform first, then add/scale” gives the same result as “add/scale first, then transform.” Equivalently, for any linear combination,

Linear example: scaling. Define $T \colon \mathbb{R}^2 \to \mathbb{R}^2$ by $T(x) = 3x$. Then

and $T(cu) = 3(cu) = c(3u) = c\, T(u)$. Scaling by a fixed factor is linear. Reflection through the origin ($T(x) = -x$), rotation about the origin, and projection onto a line through the origin are also linear — we will meet them as matrices below.

Non-linear counterexample: adding a constant. Define $S(x) = x + b$ for a fixed nonzero shift $b$ (translation). Then

These agree only when $b = 0$. Translation moves the origin; linear maps must send $0$ to $0$, because $T(0) = T(0 \cdot u) = 0 \cdot T(u) = 0$. Any map that moves the origin fails linearity.

Why this matters. Linearity is the contract that lets us represent a whole map by a finite table of numbers — a matrix — and compose maps by multiplying those tables. Everything from solving $Ax = b$ to changing bases and projecting data assumes this structure.

Matrices as Transformations#

An $m \times n$ matrix $A$ is a rectangular array of real numbers with $m$ rows and $n$ columns. Write its columns as vectors $a_1, a_2, \ldots, a_n \in \mathbb{R}^m$. For a vector $x = \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix} \in \mathbb{R}^n$, the matrix–vector product is the linear combination of those columns weighted by the entries of $x$:

The result lives in $\mathbb{R}^m$. In components, the $i$-th entry of $Ax$ is the dot product of the $i$-th row of $A$ with $x$.

Worked example. Let

The columns are $a_1 = \begin{pmatrix} 1 \\ 3 \end{pmatrix}$ and $a_2 = \begin{pmatrix} 2 \\ 4 \end{pmatrix}$, so

Linearity of $x \mapsto Ax$. By the column definition,

So every matrix defines a linear transformation $T_A(x) = Ax$ from $\mathbb{R}^n$ to $\mathbb{R}^m$.

Every linear map has a matrix. Conversely, if $T \colon \mathbb{R}^n \to \mathbb{R}^m$ is linear, let $e_1, \ldots, e_n$ be the standard basis of $\mathbb{R}^n$ and form the matrix $A$ whose $j$-th column is $T(e_j)$. Then for any $x = x_1 e_1 + \cdots + x_n e_n$,

So linear transformations $\mathbb{R}^n \to \mathbb{R}^m$ and $m \times n$ matrices are two views of the same object: the map, and the table that stores where the basis goes.

Matrix Multiplication as Composition#

If $B$ is $n \times p$ and $A$ is $m \times n$, the product $AB$ is the $m \times p$ matrix defined so that

for every $x \in \mathbb{R}^p$. In words: applying $AB$ means apply $B$ first, then apply $A$. Equivalently, $AB$ is the matrix of the composition $A \circ B$: the right-hand factor acts first, then the left-hand factor.

How to compute $AB$. The $j$-th column of $AB$ is $A$ times the $j$-th column of $B$. Equivalently, entry $(i,j)$ of $AB$ is the dot product of row $i$ of $A$ with column $j$ of $B$. The product is defined only when the inner dimensions match ($n$ above).

Numeric example. Compose two $2 \times 2$ maps. Let

Then

Check on $x = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$:

Order matters. In general $AB \neq BA$. Here

Shear then scale is not the same as scale then shear. Composition of linear maps is associative — $(AB)C = A(BC)$ — but not commutative.

Geometric Examples#

Three families of $2 \times 2$ matrices appear constantly in geometry, graphics, and robotics. Each is linear, so each is completely determined by where it sends the standard basis vectors $e_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $e_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$.

Rotation by angle $\theta$ (counterclockwise about the origin).

The first column is where $e_1$ goes: $(\cos\theta, \sin\theta)$. The second is where $e_2$ goes: $(-\sin\theta, \cos\theta)$. For $\theta = 90^\circ = \pi/2$, $\cos 90^\circ = 0$ and $\sin 90^\circ = 1$, so

This sends $(x, y)$ to $(-y, x)$: the unit square’s corner $(1,0)$ maps to $(0,1)$, and $(0,1)$ maps to $(-1,0)$. Lengths and angles are preserved; only orientation in the plane changes.

Scaling (stretch or shrink along the axes).

sends $(x, y)$ to $(ax, by)$. If $a = b$, the map is a uniform scale (similarity). If $a \neq b$, a circle becomes an ellipse elongated along the axis with larger absolute factor. For example, $\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ doubles horizontal distances and leaves vertical ones unchanged. Negative $a$ or $b$ also reflects through the corresponding axis.

Projection onto the $x$-axis.

sends $(x, y)$ to $(x, 0)$. Every point collapses vertically onto the horizontal axis. A unit square with vertices at $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$ becomes the segment from $(0,0)$ to $(1,0)$ (the top edge lands on the bottom). Projection is linear but not invertible: many inputs share the same output, and applying it twice is the same as applying it once (it is a projection operator: $P^2 = P$).

Acting on a simple shape. Take the unit square’s vertices as columns of a $2 \times 4$ matrix of points (or a closed path with five points including the return to the start). Multiplying on the left by $R_\theta$, by a diagonal scale matrix, or by the $x$-axis projection transforms every vertex the same way. The browser lab below rotates a square by $90^\circ$ and plots original versus image — the same recipe works for scaling and projection.

These geometric maps are the building blocks of Week 3’s elementary matrices (row operations as left-multiplication) and of the four-subspace picture in Week 6 (kernel and range of $A$).

Knowledge check#

Browser lab#

Apply a 90° rotation matrix to a square and plot the result.

import numpy as np
import matplotlib.pyplot as plt

theta = np.pi / 2
R = np.array([[np.cos(theta), -np.sin(theta)],
              [np.sin(theta),  np.cos(theta)]])

square = np.array([[0, 1, 1, 0, 0],
                    [0, 0, 1, 1, 0]])

rotated = R @ square

fig, ax = plt.subplots(figsize=(5, 5))
ax.plot(square[0], square[1], "o-", color="tab:blue", label="original")
ax.plot(rotated[0], rotated[1], "o-", color="tab:orange", label="rotated 90°")
ax.set_xlim(-2, 2)
ax.set_ylim(-2, 2)
ax.set_aspect("equal")
ax.axhline(0, color="gray", linewidth=0.5)
ax.axvline(0, color="gray", linewidth=0.5)
ax.legend()
ax.set_title("A square before and after applying R")
plt.show()

print("R =\n", R)