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\centerline{\bf Linear Transformations}
\bigskip

\begin{enumerate}
\item Using colored markers, draw the initial vectors, all on the same graph on your whiteboard.

$$\vert{\rm red}\rangle=1\vert \hat\imath \rangle+0\vert\hat\jmath\rangle
\doteq\pmatrix{1\cr 0\cr}\qquad
\vert{\rm green}\rangle=0\vert \hat\imath \rangle+1\vert\hat\jmath\rangle
\doteq\pmatrix{0\cr 1\cr}\qquad
\vert{\rm blue}\rangle=1\vert \hat\imath \rangle+1\vert\hat\jmath\rangle
\doteq\pmatrix{1\cr 1\cr}$$
$$\vert{\rm black}\rangle=1\vert \hat\imath \rangle-1\vert\hat\jmath\rangle
\doteq\pmatrix{1\cr -1\cr}\qquad
\vert{\rm purple}\rangle=1\vert \hat\imath \rangle+3\vert\hat\jmath\rangle
\doteq\pmatrix{1\cr 3\cr}
$$

\item  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.

$$A_1\doteq\pmatrix{0&1\cr -1&0\cr}\quad
A_2\doteq\pmatrix{0&-1\cr 1&0\cr}\quad
A_3\doteq\pmatrix{0&1\cr 1&0\cr}\quad
A_4\doteq\pmatrix{1&0\cr 0&-1\cr}$$
$$A_5\doteq\pmatrix{-1&0\cr 0&-1\cr}\quad
A_6\doteq\pmatrix{1&2\cr 1&2\cr}\quad
A_7\doteq\pmatrix{1&2\cr 9&4\cr}\quad
A_8\doteq\pmatrix{1&1\cr -1&1\cr}$$
$$A_9\doteq\pmatrix{2&0\cr 0&2\cr}\quad
A_{10}\doteq\pmatrix{1&1\cr 1&1\cr}\quad
A_{11}\doteq\pmatrix{1&0\cr 0&0\cr}\quad
A_{12}\doteq\frac{\hbar}{2}\pmatrix{1&0\cr 0&-1\cr}
$$

\item  Find the determinant of your matrix.

\item  Make note of any differences between the initial and transformed
vectors.  Specifically, look for rotations, inversions, length changes,
anything that is different.  Are there any vectors which are left unchanged by
your transformation?  Your group should be prepared to report to the
class about your transformation.

\item  When your group is done, put a sketch of your transformed vectors on
the chalkboard or prop your whitboard on a chalktray.  State what your matrix does, give the determinant of your matrix, and mention any unchanged vectors or vectors whose direction is unchanged.
\end{enumerate}

\vfill
\leftline{\it by Jason Janesky \& Corinne Manogue}
\leftline{\copyright 2011 Corinne A. Manogue}
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