[Linear Algebra] Matrix of Linear Transformation

Transformations between Vectors

• $T : \textbf{x} \in \mathbb{R}^n \mapsto \textbf{y} \in \mathbb{R}^m$ : mapping n-dim vector to m-dim vector

Matrix of Linear Transformation

• Let $T : \mathbb{R}^n \mapsto \mathbb{R}^m$ be a linear transformation, then $T$ is always written as a matrix-vector multiplication
  • $T(\textbf{x}) = A\textbf{x}$ for all $\textbf{x} \in \mathbb{R}^n$
• The j-th column of $A \in \mathbb{R}^{m \times n}$ is equal to the vector $T(\textbf{e}_j)$, where $\textbf{e}_j$ is the j-th column of the identity matrix in $\mathbb{R}^{n \times n}$
  • $A = \begin{bmatrix} T(\textbf{e}_1) \cdots T(\textbf{e}_n)\end{bmatrix}$
  • The matrix $A$ is called the standard matrix of the linear transformation $T$

e.g.

$T\begin{pmatrix} \begin{bmatrix}1 \\0 \\0 \end{bmatrix} \end{pmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}, T\begin{pmatrix} \begin{bmatrix}0 \\1 \\0 \end{bmatrix} \end{pmatrix} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}, T\begin{pmatrix} \begin{bmatrix}0 \\0 \\1 \end{bmatrix} \end{pmatrix} = \begin{bmatrix} 5 \\ 6 \end{bmatrix}$

$\textbf{x} = \begin{bmatrix} x_1 \\x_2 \\x_3 \end{bmatrix} = x_1 \begin{bmatrix} 1 \\0 \\0 \end{bmatrix} + x_2 \begin{bmatrix} 0 \\1 \\0 \end{bmatrix} + x_3 \begin{bmatrix} 0 \\0 \\1 \end{bmatrix}$

$T\textbf{x} = T\begin{pmatrix} x_1 \begin{bmatrix}1 \\0 \\0 \end{bmatrix} + x_2 \begin{bmatrix}0 \\1 \\0 \end{bmatrix} + x_3 \begin{bmatrix}0 \\0 \\1 \end{bmatrix} \end{pmatrix}$

$= x_1 T\begin{pmatrix} \begin{bmatrix}1 \\0 \\0 \end{bmatrix} \end{pmatrix} + x_2 T\begin{pmatrix} \begin{bmatrix}0 \\1 \\0 \end{bmatrix} \end{pmatrix} + x_3 T\begin{pmatrix} \begin{bmatrix}0 \\0 \\1 \end{bmatrix} \end{pmatrix}$

$= x_1 \begin{bmatrix} 1 \\ 2 \end{bmatrix} + x_2 \begin{bmatrix} 3 \\ 4 \end{bmatrix} + x_3 \begin{bmatrix} 5 \\ 6 \end{bmatrix}$

$= \begin{bmatrix} 1 & 3 & 5 \\ 2 & 4 & 6 \end{bmatrix}\begin{bmatrix} x_1 \\x_2 \\x_3 \end{bmatrix} = A\textbf{x}$

The matrix $A$ is made with standard bases, so we call them standard matrix of linear transformation $T$