Linear Algebra/Change of Basis

Representations, whether of vectors or of maps, vary with the bases. For instance, with respect to the two bases $$\mathcal{E}_2$$ and



B=\langle \begin{pmatrix} 1 \\ 1 \end{pmatrix},\begin{pmatrix} 1 \\ -1 \end{pmatrix} \rangle $$

for $$\mathbb{R}^2$$, the vector $$\vec{e}_1$$ has two different representations.



{\rm Rep}_{\mathcal{E}_2}(\vec{e}_1)=\begin{pmatrix} 1 \\ 0 \end{pmatrix} \qquad {\rm Rep}_{B}(\vec{e}_1)=\begin{pmatrix} 1/2 \\ 1/2 \end{pmatrix} $$

Similarly, with respect to $$\mathcal{E}_2,\mathcal{E}_2$$ and $$\mathcal{E}_2,B$$, the identity map has two different representations.



{\rm Rep}_{\mathcal{E}_2,\mathcal{E}_2}(\text{id})= \begin{pmatrix} 1 &0 \\ 0 &1 \end{pmatrix} \qquad {\rm Rep}_{\mathcal{E}_2,B}(\text{id})= \begin{pmatrix} 1/2 &1/2 \\ 1/2 &-1/2 \end{pmatrix} $$

With our point of view that the objects of our studies are vectors and maps, in fixing bases we are adopting a scheme of tags or names for these objects, that are convienent for computation. We will now see how to translate among these names&mdash; we will see exactly how representations vary as the bases vary.