###### 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*}
1. If the coordinates of $$\xvec$$ in the basis $$\bcal$$ are $$\coords{\xvec}{\bcal} = \twovec{-2}{4}\text{,}$$ what is the vector $$\xvec\text{?}$$

2. If $$\xvec = \twovec{3}{5}\text{,}$$ find the coordinates of $$\xvec$$ in the basis $$\bcal\text{;}$$ that is, find $$\coords{\xvec}{\bcal}\text{.}$$

3. 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.

4. 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.

5. 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 $$\bcal\text{.}$$

6. Suppose we consider the standard basis

\begin{equation*} \ecal = \{\evec_1,\evec_2\}\text{.} \end{equation*}

What is the relationship between $$\xvec$$ and $$\coords{\xvec}{\ecal}\text{?}$$

in-context