User:IssaRice/Linear algebra/Change of basis example in two dimensions

This example comes from this video. To make it easier to go back and forth between this page and the video, the notation on this page follows that of the video.

We are working in $R^{2}$ , the plane.

Jennifer's basis vectors: ${\vec{b}}_{1} : = [\begin{matrix} 2 \\ 1 \end{matrix}]$ and ${\vec{b}}_{2} : = [\begin{matrix} - 1 \\ 1 \end{matrix}]$

To Jennifer, ${\vec{b}}_{1}$ looks like $[\begin{matrix} 1 \\ 0 \end{matrix}]$ and ${\vec{b}}_{2}$ looks like $[\begin{matrix} 0 \\ 1 \end{matrix}]$ .

If Jennifer says " $[\begin{matrix} - 1 \\ 2 \end{matrix}]$ ", to us (in the standard basis) this is the vector $- 1 {\vec{b}}_{1} + 2 {\vec{b}}_{2} = - 1 [\begin{matrix} 2 \\ 1 \end{matrix}] + 2 [\begin{matrix} - 1 \\ 1 \end{matrix}] = [\begin{matrix} - 4 \\ 1 \end{matrix}]$ .

We can also write the above calculation as $[\begin{matrix} {\vec{b}}_{1} & {\vec{b}}_{2} \end{matrix}] [\begin{matrix} - 1 \\ 2 \end{matrix}] = [\begin{matrix} 2 & - 1 \\ 1 & 1 \end{matrix}] [\begin{matrix} - 1 \\ 2 \end{matrix}] = [\begin{matrix} - 4 \\ 1 \end{matrix}]$ .

Notice that $[\begin{matrix} {\vec{b}}_{1} & {\vec{b}}_{2} \end{matrix}] {\vec{e}}_{1} = {\vec{b}}_{1}$ and $[\begin{matrix} {\vec{b}}_{1} & {\vec{b}}_{2} \end{matrix}] {\vec{e}}_{2} = {\vec{b}}_{2}$ , i.e., this matrix transforms our (standard) basis vectors into Jennifer's basis vectors.