1. Many linear transformations are secretly diagonal

Many linear transformations are like diagonal transformations if you look at them the right way--they are ``diagonalizable''. If the matrix is not already diagonal, we need to choose a new basis, representing new axes.

Figure shows the effects of the transformation $\tau _ A$ for $A = \matp{rr}{4&1\\ 1&4}$. But instead of showing the usual unit square and its image (a parallelogram), the figure shows a special slanted square and its image. Also shown are the vectors v$_ 1$, v$_ 2$ along the square, and their images.

Figure: A diagonalizable transformation

udir/secret.eps

Problem U-2. Find $\tau _ A ($v$_ 1)$ and $\tau _ A ($v$_ 2)$ numerically and see how they relate to v$_ 1$ and v$_ 2$. Here v$_ 1 = \matp{r}{1\\ 1}$ and v$_ 2 = \matp{r}{-1\\ 1}$.

Problem U-3. Show that the matrix of $\tau _ A$ relative to the new basis v$_ 1$, v$_ 2$ is a diagonal matrix, specifically, the diagonal matrix $D = \matp{rr}{5&0\\ 0&3}$.