###### Exercise2

Consider vectors

\begin{equation*} \begin{aligned} \vvec_1=\twovec{1}{2}, \amp \vvec_2=\twovec{1}{-3} \\ \wvec_1=\twovec{2}{3}, \amp \wvec_2=\twovec{-1}{-2} \text{.} \\ \end{aligned} \end{equation*}

and let $$\bcal = \{\vvec_1,\vvec_2\}$$ and $$\ccal = \{\wvec_1,\wvec_2\}\text{.}$$

1. Explain why $$\bcal$$ and $$\ccal$$ are both bases of $$\real^2\text{.}$$

2. If $$\xvec = \twovec{5}{-3}\text{,}$$ find $$\coords{\xvec}{\bcal}$$ and $$\coords{\xvec}{\ccal}\text{.}$$

3. If $$\coords{\xvec}{\bcal}=\twovec{2}{-4}\text{,}$$ find $$\xvec$$ and $$\coords{\xvec}{\ccal}\text{.}$$

4. If $$\coords{\xvec}{\ccal}=\twovec{-3}{2}\text{,}$$ find $$\xvec$$ and $$\coords{\xvec}{\bcal}\text{.}$$

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

in-context