Exercise 9

Suppose that $A=\left[\begin{array}{rr} 2 & 1 \\ 1& 2 \\ \end{array}\right]$ and $$\vvec_1=\twovec{1}{1},~~~ \vvec_2=\twovec{1}{-1}$$.

1. Explain why $\bcal=\{\vvec_1,\vvec_2\}$ is a basis for $\real^2$.

2. Find $A\vvec_1$ and $A\vvec_2$.

3. Use what you found in the previous part of this problem to find $\coords{A\vvec_1}{\bcal}$ and $\coords{A\vvec_2}{\bcal}$.

4. If $\coords{\xvec}{\bcal} = \twovec{1}{-5}$, find $\coords{A\xvec}{\bcal}$.

5. Find a matrix $D$ such that $\coords{A\xvec}{\bcal} = D\coords{\xvec}{\bcal}$.

You should find that the matrix $D$ is a very simple matrix, which means that this basis $\bcal$ is well suited to study the effect of multiplication by $A$. This observation is the central idea of the next chapter.