###### Exercise9

Suppose that $$A=\left[\begin{array}{rr} 2 \amp 1 \\ 1\amp 2 \\ \end{array}\right]$$ and

\begin{equation*} \vvec_1=\twovec{1}{1}, \vvec_2=\twovec{1}{-1}\text{.} \end{equation*}
1. Explain why $$\bcal=\{\vvec_1,\vvec_2\}$$ is a basis for $$\real^2\text{.}$$

2. Find $$A\vvec_1$$ and $$A\vvec_2\text{.}$$

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

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

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

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 $$A\text{.}$$ This observation is the central idea of the next chapter.

in-context