Linear Algebra/Changing Representations of Vectors

Template:Navigation In converting ${\mathrm{R}\mathrm{e}\mathrm{p}}_B(\overrightarrow{v})$ to ${\mathrm{R}\mathrm{e}\mathrm{p}}_D(\overrightarrow{v})$ the underlying vector $\overrightarrow{v}$ doesn't change.Template:Anchor Thus, this translation is accomplished by the identity map on the space, described so that the domain space vectors are represented with respect to $B$ and the codomain space vectors are represented with respect to $D$.

(The diagram is vertical to fit with the ones in the next subsection.)

We finish this subsection by recognizing that the change of basis matrices are familiar.

In the next subsection we will see how to translate among representations of maps, that is, how to change ${\mathrm{R}\mathrm{e}\mathrm{p}}_{B,D}(h)$ to ${\mathrm{R}\mathrm{e}\mathrm{p}}_{\hat{B},\hat{D}}(h)$. The above corollary is a special case of this, where the domain and range are the same space, and where the map is the identity map.

Exercises

/Solutions/

For more exercises see Abstract Algebra/3x3 real matrices.