Change of Basis

Theorem

• With $V$ as a finite dimensional vector space
• With $\alpha ,\beta$ being finite dimensional basis of $V$

$[I{]}_{\alpha }^{\beta }[v{]}_{\alpha }=[v{]}_{\beta }$ Where $[I{]}_{\alpha }^{\beta }$ is the change of basis matrix

Computing $[I{]}_{\alpha }^{\beta }$ Process

1. Input the basis vectors of $\alpha$
2. Write all outputs in basis $\beta$

Example

Consider bases $\alpha =\{(0,1),(1,0)\}$ and $\beta =\{(1,1),(1,-1)\}$ Write matrix $[T{]}_{\alpha }^{\beta }$ of the identity transformation $I:{\mathbb{R}}^2\to {\mathbb{R}}^2$ . Calculate:

• $T(0,1)=\frac{1}{2}(1,1)-\frac{1}{2}(1,-1)$
• $T(1,0)=\frac{1}{2}(1.1)+\frac{1}{2}(1,-1)$ This gives:

$$[T{]}_{\alpha }^{\beta }\left[\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ -\frac{1}{2}& \frac{1}{2}\end{array}\right]$$