Exercise 2

Consider vectors $$\begin{aligned} \vvec_1=\twovec{1}{2}, & \vvec_2=\twovec{1}{-3} \\ \wvec_1=\twovec{2}{3}, & \wvec_2=\twovec{-1}{-2} \text{.} \\ \end{aligned}$$ and let $\bcal = \{\vvec_1,\vvec_2\}$ and $\ccal = \{\wvec_1,\wvec_2\}$.

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

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

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

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

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