Activity3.2.3
Let's begin with the basis \(\bcal = \{\vvec_1,\vvec_2\}\) of \(\real^2\) where
\begin{equation*} \vvec_1 = \twovec{3}{2}, \vvec_2 = \twovec{2}{1} \text{.} \end{equation*}If the coordinates of \(\xvec\) in the basis \(\bcal\) are \(\coords{\xvec}{\bcal} = \twovec{2}{4}\text{,}\) what is the vector \(\xvec\text{?}\)
If \(\xvec = \twovec{3}{5}\text{,}\) find the coordinates of \(\xvec\) in the basis \(\bcal\text{;}\) that is, find \(\coords{\xvec}{\bcal}\text{.}\)
Find a matrix \(A\) such that, for any vector \(\xvec\text{,}\) we have \(\xvec = A\coords{\xvec}{\bcal}\text{.}\) Explain why this matrix is invertible.
Using what you found in the previous part, find a matrix \(B\) such that, for any vector \(\xvec\text{,}\) we have \(\coords{\xvec}{\bcal} = B\xvec\text{.}\) What is the relationship between the two matrices you have found in this and the previous part? Explain why this relationship holds.

Suppose we also consider the basis
\begin{equation*} \ccal = \left\{\twovec{1}{2}, \twovec{2}{1}\right\} \text{.} \end{equation*}Find a matrix \(C\) that converts coordinates in the basis \(\ccal\) into coordinates in the basis \(\bcal\text{;}\) that is,
\begin{equation*} \coords{\xvec}{\bcal} = C \coords{\xvec}{\ccal} \text{.} \end{equation*}You may wish to think about converting coordinates from the basis \(\ccal\) into the standard coordinate system and then into the basis \(\bcal\text{.}\)

Suppose we consider the standard basis
\begin{equation*} \ecal = \{\evec_1,\evec_2\}\text{.} \end{equation*}What is the relationship between \(\xvec\) and \(\coords{\xvec}{\ecal}\text{?}\)