Matrix Representation

58. Matrix Representation

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

𝑇 (\vec{𝑥}) = 𝐴 \vec{𝑥}

Conversely, every $𝑚 \times 𝑛$ matrix gives a linear transformation. So linear transformations $ℝ^{𝑛} \to ℝ^{𝑚}$ are the same as $𝑚 \times 𝑛$ matrices.

58.1. How to find the matrix

Apply $𝑇$ to each standard basis vector ${\vec{𝑒}}_{1}, \dots, {\vec{𝑒}}_{𝑛}$ . The results become the columns of $𝐴$ :

𝐴 = [\begin{matrix} 𝑇 ({\vec{𝑒}}_{1}) & 𝑇 ({\vec{𝑒}}_{2}) & \dots & 𝑇 ({\vec{𝑒}}_{𝑛}) \end{matrix}]

58.2. Why this works (sketch)

Any input $\vec{𝑥} \in ℝ^{𝑛}$ is a linear combination of standard basis vectors:

\vec{𝑥} = 𝑥_{1} {\vec{𝑒}}_{1} + 𝑥_{2} {\vec{𝑒}}_{2} + \dots + 𝑥_{𝑛} {\vec{𝑒}}_{𝑛}

Applying $𝑇$ and using linearity:

\begin{aligned} 𝑇 (\vec{𝑥}) & = 𝑇 (𝑥_{1} {\vec{𝑒}}_{1} + \dots + 𝑥_{𝑛} {\vec{𝑒}}_{𝑛}) \\ = 𝑥_{1} 𝑇 ({\vec{𝑒}}_{1}) + \dots + 𝑥_{𝑛} 𝑇 ({\vec{𝑒}}_{𝑛}) \\ = [\begin{matrix} 𝑇 ({\vec{𝑒}}_{1}) & \dots & 𝑇 ({\vec{𝑒}}_{𝑛}) \end{matrix}] \vec{𝑥} \\ = 𝐴 \vec{𝑥} \end{aligned}

The columns of $𝐴$ are exactly the images of the standard basis under $𝑇$ .

Example

$𝑇 : ℝ^{2} \to ℝ^{2}$ rotates by 90° counterclockwise.

$𝑇 ({\vec{𝑒}}_{1}) = 𝑇 ([\begin{matrix} 1 \\ 0 \end{matrix}]) = [\begin{matrix} 0 \\ 1 \end{matrix}]$ , $𝑇 ({\vec{𝑒}}_{2}) = 𝑇 ([\begin{matrix} 0 \\ 1 \end{matrix}]) = [\begin{matrix} - 1 \\ 0 \end{matrix}]$ .

𝐴 = [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}]

Verify: $𝐴 [\begin{matrix} 2 \\ 3 \end{matrix}] = [\begin{matrix} - 3 \\ 2 \end{matrix}]$ — rotating $(2, 3)$ by 90°.

58.3. Identity transformation

The identity $𝐼 : ℝ^{𝑛} \to ℝ^{𝑛}$ , $𝐼 (\vec{𝑥}) = \vec{𝑥}$ , has matrix $𝐼_{𝑛}$ (the identity matrix), since $𝐼 ({\vec{𝑒}}_{𝑖}) = {\vec{𝑒}}_{𝑖}$ for each standard basis vector.

58.4. See also

Linear Transformation — definition
Matrix–Vector Product — the operation that makes a matrix into a transformation
Change of Basis — what happens to the matrix when you switch bases
Rotation Matrix, Reflection Matrix, Scaling Matrix, Shear Matrix — common examples