Week 2: Linear Transformations and Matrices

What Is a Linear Transformation#

A map from $\mathbb{R}^n$ to $\mathbb{R}^m$ is normally unknowable: to know one, you would have to inspect where every single point goes. The linear transformations (or linear maps) are the maps that respect the two vector-space operations, addition and scalar multiplication — a single constraint that is enough to store the whole map as a finite table of numbers. Formally, $T$ is linear when both of the following hold for all vectors $u, v$ and every scalar $c$:

Together these say the map can be pulled through either operation in either order: “transform first, then add or scale” gives the same result as “add or scale first, then transform.” 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 $b = 0$. The obstruction is the origin: a linear map must fix $0$, since $T(0) = T(0 \cdot u) = 0 \cdot T(u) = 0$, so any translation by a nonzero vector fails linearity.

Why this matters. Linearity is the contract that makes composition cheap. Once a map is stored as a matrix, applying it to any input is one matrix–vector product, and composing two maps is one matrix multiplication. Everything from solving $Ax = b$ to changing bases and projecting data reduces to these operations.

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 $A$ — the right-hand factor acts first and its output feeds the left-hand factor. So $AB$ is exactly the matrix of the composition $A \circ B$.

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, as function composition always is, so $(AB)C = A(BC)$ — but it is 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 $(-1,0)$. Rotation is a rigid motion: $\|R_\theta (x, y)\|^2 = (x\cos\theta - y\sin\theta)^2 + (x\sin\theta + y\cos\theta)^2 = x^2 + y^2$, the cross terms cancelling — so lengths, and therefore the angles between vectors, are unchanged.

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 $A$). Rotation, scaling, and projection matrices are also the working language of rigid-body motion in Robot Learning Week 1: Robot Modeling and Kinematics.

The unit square (dashed) and its image under the shear $A=\begin{pmatrix}1&1\\0&1\end{pmatrix}$. The columns of $A$ are where $e_1$ and $e_2$ land, so the image is the parallelogram spanned by those columns.

Knowledge check#

Browser lab: rotation matrix on a square#

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)

print("Notice: R ≈ [[0, -1], [1, 0]] — the square keeps its shape and every corner rotates 90° counter-clockwise.")

Done when#

You can state the linear transformation a given 2×2 matrix represents, compute its action on a vector, and multiply two matrices as composed transformations. Check it against the browser lab: the printed R is the 90° rotation $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$, the plot shows the square rotated a quarter-turn counter-clockwise, and $R(1,0)^\top = (0,1)^\top$ matches your hand computation.

Further Reading#

• Gilbert Strang, Introduction to Linear Algebra (5th ed., Wellesley-Cambridge Press, 2016). Chapter 2 covers linear transformations and matrix multiplication, with the "matrix as a function" viewpoint.
• David C. Lay, Steven R. Lay & Judi J. McDonald, Linear Algebra and Its Applications (6th ed., Pearson, 2021). §§1.8–1.9 introduce linear transformations and their standard matrices.
• 3Blue1Brown, Essence of Linear Algebra. The videos "Linear transformations and matrices", "Matrix multiplication as composition", and "Three-dimensional linear transformations" are the geometric companion to this week.
• Stephen Boyd & Lieven Vandenberghe, Introduction to Applied Linear Algebra (Cambridge University Press, 2018). Chapter 6 treats linear transformations as matrices from an applied perspective. Freely available at vmls-book.stanford.edu.