Change of Basis Matrix

Let $\mathcal{A}$ and $\mathcal{B}$ be bases for $\R^n$, matrix $M$ is a change of basis matrix if:

$M[\vec{x}]{\mathcal{A}}=[\vec{x}]{\mathcal{B}} ~~~~ \forall \vec{x}\in\R^n$

$M=[\mathcal{B} \leftarrow \mathcal{A}]$

Note: Change of basis matrix are one-way.

Change of Basis Matrix Properties

• $[C\leftarrow A]=[C \leftarrow B][B \leftarrow A]$
• $[B \leftarrow A] = [A \leftarrow B]^{-1}$
• An $n \times n$ matrix $M$ is invertible $\Leftrightarrow M$ is a change of basis matrix

Finding Change of Basis Matrices

Let $A=\{\vec{a}_1, \vec{a}_2\}, B=\{\vec{b}_1, \vec{b}_2\}$

1. Find $\vec{x}_1=[\vec{a}_1]_B,\vec{x}_2=[\vec{a}_2]_B$
2. $[B \leftarrow A]=\begin{bmatrix} \vec{x}_1 & \vec{x}_2 \end{bmatrix}$

Linear Transformation in a Basis

Let $T:\R^n\to\R^n$ be a linear transformation and let $\mathcal{B}$ be a basis for $\R^n$

The matrix for $T$ with respect to $\mathcal{B}$ is $[T]_{\mathcal{B}}$

$[T(\vec{x})]{\mathcal{B}}=[T]{\mathcal{B}}[\vec{x}]_{\mathcal{B}}$

$M=[T_M]_\mathcal{E}$

$[T]B=[B\leftarrow \mathcal{E}][T]\mathcal{E}[\mathcal{E} \leftarrow B]$

Similar Matrices