Students explore what linear transformation matrices do to vectors. The whole class discussion compares & contrasts several different types of transformations (rotation, flip, projections, “scrinch”, scale) and how the properties of the matrices (the determinant, symmetries, which vectors are unchanged) are related to these transformations.
Using colored markers, draw these initial vectors, all on the same graph on your whiteboard.
\[\vec{v}_{red}=1\vert \hat x \rangle+0\vert\hat y\rangle \doteq\begin{pmatrix}1\\ 0\end{pmatrix}\qquad \vec{v}_{green}=0\vert \hat x \rangle+1\vert\hat y\rangle \doteq\begin{pmatrix}0\\ 1\end{pmatrix}\qquad \vec{v}_{blue}=1\vert \hat x \rangle+1\vert\hat y\rangle \doteq\begin{pmatrix}1\\ 1\end{pmatrix}\] \[\vec{v}_{black}=1\vert \hat x \rangle-1\vert\hat y\rangle \doteq\begin{pmatrix}1\\ -1\end{pmatrix}\qquad \vec{v}_{purple}=1\vert \hat x \rangle+3\vert\hat y\rangle \doteq\begin{pmatrix}1\\ 3\end{pmatrix} \]
Each group will be assigned one of the following matrices. Operate on the initial vectors with your group's matrix and graph the transformed vectors on a single (new) graph.
\[A_1\doteq\begin{pmatrix}0&1\\ -1&0\end{pmatrix}\quad A_2\doteq\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}\quad A_3\doteq\begin{pmatrix}0&1\\ 1&0\end{pmatrix}\quad A_4\doteq\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}\] \[A_5\doteq\begin{pmatrix}-1&0\\ 0&-1\end{pmatrix}\quad A_6\doteq\begin{pmatrix}1&2\\ 1&2\end{pmatrix}\quad A_7\doteq\begin{pmatrix}1&2\\ 9&4\end{pmatrix}\quad A_8\doteq\begin{pmatrix}1&1\\ -1&1\end{pmatrix}\] \[A_9\doteq\begin{pmatrix}2&0\\ 0&2\end{pmatrix}\quad A_{10}\doteq\begin{pmatrix}1&1\\ 1&1\end{pmatrix}\quad A_{11}\doteq\begin{pmatrix}1&0\\ 0&0\end{pmatrix}\quad A_{12}\doteq\frac{\hbar}{2}\begin{pmatrix}1&0\\ 0&-1\end{pmatrix} \]