How to find the matrix of a transformation

Question

From the wikipedia page about transformation $T$:

$$T : \mathbb{R}^n \to \mathbb{R}^m$$

$\vec{x}$ is a column vector with $n$ entries.

$$T(\vec{x}) = \textbf{A}\vec{x}$$

$\textbf{A}$ is the transformation matrix of $T$.

It is easy to determine the transformation matrix $\textbf{A}$ by transforming each of the vectors of the standard basis by $T$, then inserting the result into the columns of a matrix:

$$\mathbf{A} = \begin{bmatrix} T( \vec e_1 ) & T( \vec e_2 ) & \cdots & T( \vec e_n ) \end{bmatrix}$$

Example: $T(x) = 5x$ is a linear transformation.

Applying the above process (suppose that n = 2 in this case) reveals that:

$$T( \vec{x} ) = 5 \vec{x} = 5 \mathbf{I} \vec{x} = \begin{bmatrix} 5 && 0 \\ 0 && 5 \end{bmatrix} \vec{x}$$

The questions are:

1. What is "each of the vectors of the standard basis".
2. How does the last equation work (wondering why the matrix is 2x2, and what the $\textbf{I}$ is, etc.) $$T( \vec{x} ) = 5 \vec{x} = 5 \mathbf{I} \vec{x} = \begin{bmatrix} 5 && 0 \\ 0 && 5 \end{bmatrix} \vec{x}$$
3. Are they always square matrices? Wondering the general strategy for constructing a transformation matrix.

Answer 1

1. What is "each of the vectors of the standard basis".

The vectors of the standard basis are

• $e_1=(1,0,...,0) $
• $e_2=(0,1,...,0) $
• ...
• $e_n=(0,0,...,1) $

2. How does the last equation work (wondering why the matrix is 2x2, and what the $\textbf{I}$ is, etc.) $$T( \vec{x} ) = 5 \vec{x} = 5 \mathbf{I} \vec{x} = \begin{bmatrix} 5 && 0 \\ 0 && 5 \end{bmatrix} \vec{x}$$

In this case we are looking for $T : \mathbb{R}^2 \to \mathbb{R}^2$ and $\mathbf{I}$ is the identity matrix $\begin{bmatrix} 1 && 0 \\ 0 && 1 \end{bmatrix}$ of order $2$.

3. Are they always square matrices? Wondering the general strategy for constructing a transformation matrix.

In general for $T : \mathbb{R}^n \to \mathbb{R}^m$ the dimension of the matrix which represents the linear transformation is n-by-m that is with $n$ columns and $m$ rows.

For the construction, with respect to a given basis, the general strategy is to determine the transformed vectors for the vectors of the given basis. That is, if we refer to the standard basis

$$\mathbf{A} = \begin{bmatrix} T( \vec e_1 ) & T( \vec e_2 ) & \cdots & T( \vec e_n ) \end{bmatrix}$$

Note that more in general we could also choose different basis, and not necessarly the standard one among them, for $\vec x\in\mathbb{R}^n$ and $T[\vec x]\in\mathbb{R}^m$.

Answer 2

• The standard basis of the Euclidean space of $\mathbb{R}^n$ is $\{\vec{e}_1, \ldots, \vec{e}_n\}$ where $\vec{e}_i$ is the vector with $1$ in its $i$-th component and $0$ elsewhere.

• $I$ refers to the identity matrix. In the example, $n=2$, $I=\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$, it is a diagonal matrix with $1$ along the diagonal.

• A particular example $T: \mathbb{R}^2 \to \mathbb{R}^2$ is shown with the rules of $T(x)=5x$. $$T \left( \vec{e}_1 \right)=5\vec{e}_1, T \left( \vec{e}_2 \right)=5\vec{e}_2$$

hence, the transformation matrix is

$$\begin{bmatrix} 5\vec{e}_1 & 5 \vec{e}_2\end{bmatrix}= \begin{bmatrix} 5 & 0 \\ 0 & 5\end{bmatrix}$$

• It need not be a square matrix in general, for $T: \mathbb{R}^n \to \mathbb{R}^m$, the matrix is of size $m \times n$.

• The general strategy has been given by the wikipedia page.

The transformation matrix is

$$\mathbf{A} = \begin{bmatrix} T( \vec e_1 ) & T( \vec e_2 ) & \cdots & T( \vec e_n ) \end{bmatrix}$$

Answer 3

1) What is "each of the vectors of the standard basis".

The vectors in standard basis are $\begin {pmatrix} 1\\0\\0\\.\\.\\.\end{pmatrix}$, $\begin {pmatrix} 0\\1\\0\\.\\.\\.\end{pmatrix}$,$\begin {pmatrix} 0\\0\\1\\.\\.\\.\end{pmatrix}$,...,$\begin {pmatrix} 0\\0\\0\\.\\.\\1\end{pmatrix}$

2) How does the last equation work (wondering why the matrix is 2x2, and what the I is, etc.)$$T( \vec{x} ) = 5 \vec{x} = 5 \mathbf{I} \vec{x} = \begin{bmatrix} 5 && 0 \\ 0 && 5 \end{bmatrix} \vec{x}$$

This is just an example in two dimensional space $\mathbb {R}^2$ where $x$ is a column vector and $I$ is the identity matrix.

3) Are they always square matrices? Wondering the general strategy for constructing a transformation matrix. No, It depends on your transformation. The transformation matrices are not necessarily square matrices.