Lesson 14.17 — Matrix Multiplication & Linear Transformations

1 Introduction: Executing Many Calculations with One Operation

Suppose you are developing a 2D video game. A character currently stands at position $(2, 3)$. You want to rotate this character 90 degrees counterclockwise around the coordinate origin.

What is the character's new position? Calculating sines and cosines individually for every single vertex in an animation model would be tedious and computationally inefficient. In modern Computer Graphics and game engines, this entire spatial rotation is executed in a single matrix-vector multiplication:

Instantly, the new position $(-3, 2)$ is computed! Without manually evaluating trigonometric identities, a matrix transformation has mapped the coordinates cleanly.

2 What is Matrix Multiplication?

Unlike scalar arithmetic, matrix multiplication is not an element-by-element operation. Instead, it computes the dot products between the rows of the first matrix and the columns of the second matrix.

Consider two general $2 \times 2$ matrices:

Their product $C = AB$ is:

To compute entry $c_{ij}$, you take the dot product of Row $i$ of matrix $A$ and Column $j$ of matrix $B$.

3 Dimension Compatibility Rules

Matrix multiplication is only defined under a strict dimensional constraint:

4 Step-by-Step Worked Example

Let us work through a complete concrete computation. Given:

Because $A$ has 3 columns and $B$ has 3 rows, their product $C = AB$ is a $2 \times 2$ matrix:

5 General Mathematical Formula

For any $m \times n$ matrix $A$ and $n \times p$ matrix $B$, the general summation formula for entry $c_{ij}$ in product $C = AB$ is:

Here, $c_{ij}$ represents the entry at row $i$ and column $j$ in the product matrix, $a_{ik}$ iterates across Row $i$ of $A$, and $b_{kj}$ iterates down Column $j$ of $B$. The index $k$ runs from $1$ to $n$.

6 Why Matrix Multiplication is Non-Commutative

In ordinary arithmetic, multiplication is commutative: $2 \times 3 = 3 \times 2 = 6$. For matrices, however, changing the order changes the result completely:

Let us prove this numerically with two simple $2 \times 2$ matrices:

$AB$ and $BA$ are completely different matrices! Moreover, for non-square matrices (e.g., $2 \times 3$ and $3 \times 4$), $AB$ exists while $BA$ cannot even be evaluated.

7 Core Algebraic Properties of Matrix Multiplication

8 Matrix–Vector Multiplication

Multiplying an $m \times n$ matrix $A$ by an $n \times 1$ column vector $\mathbf{x}$ produces an $m \times 1$ output vector:

Here, matrix $A$ acts as a transformation engine that maps input vector $\begin{bmatrix} 4 \\ 2 \end{bmatrix}$ to output vector $\begin{bmatrix} 10 \\ 10 \end{bmatrix}$.

9 What is a Linear Transformation?

A Linear Transformation $T$ is a vector space mapping that strictly preserves two core mathematical axioms:

10 How a Matrix Represents a Linear Transformation

Every finite-dimensional linear transformation can be expressed as a matrix multiplication:

How does this work geometrically? In 2D Cartesian space, we have standard basis vectors:

Where $\hat{\mathbf{i}}$ lands after transformation forms the first column of $A$, and where $\hat{\mathbf{j}}$ lands forms the second column:

11 Core 2D Linear Transformations

12 Linear Transformations vs. Translation

A fundamental distinction in computer graphics: Is translating (moving) an object a Linear Transformation?

Answer: No! Ordinary translation is not linear.

Linear transformations must keep the origin invariant ($T(\mathbf{0}) = \mathbf{0}$). A translation $T(x, y) = (x+a, y+b)$ maps $(0, 0)$ to $(a, b) \ne (0, 0)$.

13 Chaining Composite Transformations

In real game engines, an object may be scaled ($S$), then rotated ($R$). Matrix multiplication allows chaining these into a single Composite Transformation Matrix:

14 Matrix Multiplication & Transformations in Python (NumPy)

Use the @ operator or np.matmul() for true matrix multiplication. Remember that * executes element-wise multiplication!

import numpy as np

# 1. Matrix multiplication with @ operator
A = np.array([[1, 2, 3], [4, 5, 6]])
B = np.array([[7, 8], [9, 10], [11, 12]])
C = A @ B  # Shape: (2, 2)
print("A @ B:\n", C)

# 2. 2D rotation matrix (90 degrees = pi/2)
theta = np.pi / 2
R = np.array([
    [np.cos(theta), -np.sin(theta)],
    [np.sin(theta),  np.cos(theta)]
])

pos = np.array([2, 3])
rotated_pos = R @ pos
print("Rotated Position:", np.round(rotated_pos))  # [-3, 2]

# 3. Composite transformation (Scale then Rotate)
S = np.array([[2, 0], [0, 3]])
composite = R @ S
print("Composite Transform:", composite @ pos)

15 Real-World Applications in Computer Science & AI

16 8 Common Pitfalls

17 Key Takeaways