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