The Matrix of a Linear Transformation

The matrix of a linear transformation provides a concise representation of how the transformation acts on vectors in a vector space. By representing a linear transformation with a matrix, we can use matrix operations to analyze and compute its effects.

Definition

A linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ can be represented by an $m \times n$ matrix $A$ , such that:

$T(\mathbf{x}) = A\mathbf{x},$

where $\mathbf{x}$ is a vector in $\mathbb{R}^n$ , and $T(\mathbf{x})$ is the transformed vector in $\mathbb{R}^m$ .

Finding the Matrix of a Linear Transformation

To find the matrix $A$ of a linear transformation $T$ :

Apply $T$ to each standard basis vector of $\mathbb{R}^n$ :

$T(\mathbf{e}_1), T(\mathbf{e}_2), \dots, T(\mathbf{e}_n).$

Use the resulting vectors as the columns of the matrix $A$ :

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

Example

Consider the linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ defined by:

$T(x, y) = (2x + y, x - 3y).$

To find the matrix $A$ :

Apply $T$ to the standard basis vectors $\mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ and $\mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$ :

$T(\mathbf{e}_1) = T\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 2(1) + 0 \\ 1 - 3(0) \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}.$
$T(\mathbf{e}_2) = T\begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 2(0) + 1 \\ 0 - 3(1) \end{bmatrix} = \begin{bmatrix} 1 \\ -3 \end{bmatrix}.$

Form the matrix $A$ using $T(\mathbf{e}_1)$ and $T(\mathbf{e}_2)$ as columns:

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

Key Properties

Uniqueness: The matrix $A$ is unique for a given linear transformation $T$ and basis.
Composition: If $T_1$ and $T_2$ are linear transformations with matrices $A_1$ and $A_2$ , the composition $T_2(T_1(\mathbf{x}))$ is represented by the matrix product:
$A_2A_1.$
Change of Basis: The matrix of $T$ changes when expressed relative to a different basis. The relationship is given by:
$A' = P^{-1}AP,$
where $P$ is the change-of-basis matrix.

Applications

Understanding the matrix of a linear transformation is crucial in many applications:

Computer Graphics: Scaling, rotation, reflection, and projection transformations are represented as matrices.
Physics: Linear transformations describe rotations and scaling in mechanics and quantum systems.
Machine Learning: Transforming datasets using matrices to analyze and preprocess data.
Engineering: Representing and solving systems of linear equations in structural analysis and control systems.

Conclusion

The matrix of a linear transformation provides a powerful framework for analyzing and computing the effects of transformations. By understanding how to construct and interpret these matrices, we can solve complex problems in a wide range of disciplines.