Matrix Representation

Overview

Every linear transformation between finite-dimensional vector spaces can be represented by a matrix. This correspondence is fundamental to linear algebra and connects abstract transformations to concrete calculations.

The Standard Matrix

For $T: \mathbb{R}^n \to \mathbb{R}^m$, the standard matrix $A$ satisfies:

$T(\mathbf{x}) = A\mathbf{x} \quad \text{for all } \mathbf{x} \in \mathbb{R}^n$

Finding the Standard Matrix

The columns of $A$ are the images of the standard basis vectors:

$A = \begin{bmatrix} T(\mathbf{e}_1) & T(\mathbf{e}_2) & \cdots & T(\mathbf{e}_n) \end{bmatrix}$

where $\mathbf{e}_i = [0, \ldots, 1, \ldots, 0]^T$ (1 in position $i$).

Example: Finding Standard Matrix

$T: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $T(x, y) = (2x + y, x - 3y)$

$T(\mathbf{e}_1) = T(1, 0) = (2, 1)$ $T(\mathbf{e}_2) = T(0, 1) = (1, -3)$

$A = \begin{bmatrix} 2 & 1 \\ 1 & -3 \end{bmatrix}$

Verify: $T(x, y) = \begin{bmatrix} 2 & 1 \\ 1 & -3 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} = (2x + y, x - 3y)$ ✓

Common Transformation Matrices

Rotation (2D) by $\theta$

$R_\theta = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$

Scaling

$S = \begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix}$

Reflection Across x-axis

$M = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

Reflection Across y-axis

$M = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$

Projection onto x-axis

$P = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$

Shear

Horizontal shear:

$\begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}$

Change of Basis

If $B = \{\mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_n\}$ is a basis for $V$, the matrix representation changes.

Coordinate Vector

For $\mathbf{v} \in V$ with $\mathbf{v} = c_1\mathbf{b}_1 + c_2\mathbf{b}_2 + \cdots + c_n\mathbf{b}_n$:

$[\mathbf{v}]_B = \begin{bmatrix} c_1 \\ c_2 \\ \vdots \\ c_n \end{bmatrix}$

Change of Basis Matrix

From basis $B$ to standard basis:

$P = \begin{bmatrix} \mathbf{b}_1 & \mathbf{b}_2 & \cdots & \mathbf{b}_n \end{bmatrix}$

Then:

$\mathbf{v} = P[\mathbf{v}]_B$

$[\mathbf{v}]_B = P^{-1}\mathbf{v}$

Transformation in Different Basis

If $T$ has matrix $A$ in standard basis and $B$ is another basis:

$[T]_B = P^{-1}AP$

Example: Change of Basis

$B = \left\{\begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ -1 \end{bmatrix}\right\}$ is a basis for $\mathbb{R}^2$

$P = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$

$P^{-1} = \frac{1}{2}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$

For $T$ with standard matrix $A = \begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix}$:

$[T]_B = P^{-1}AP = \frac{1}{2}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}\begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$

Similarity

Matrices $A$ and $B$ are similar if:

$B = P^{-1}AP \quad \text{for some invertible } P$

Similar matrices:

• Represent the same transformation in different bases
• Have the same eigenvalues
• Have the same determinant
• Have the same trace
• Have the same rank

Summary

Transformation	Matrix
Rotation by $\theta$	$\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$
Scale by $(a, b)$	$\begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}$
Reflect x-axis	$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$
Reflect y-axis	$\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$
Reflect $y = x$	$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
Project onto x	$\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$