Activity 3.2.3

Let's begin with the basis $\bcal = \{\vvec_1,\vvec_2\}$ of $\real^2$ where $$\vvec_1 = \twovec{3}{-2}, \vvec_2 = \twovec{2}{1}$$.

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

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

3. Find a matrix $A$ such that, for any vector $\xvec$, we have $\xvec = A\coords{\xvec}{\bcal}$. Explain why this matrix is invertible.

4. Using what you found in the previous part, find a matrix $B$ such that, for any vector $\xvec$, we have $\coords{\xvec}{\bcal} = B\xvec$. What is the relationship between the two matrices $A$ and $B$? Explain why this relationship holds.

5. Suppose we consider the standard basis $$\ecal = \{\evec_1,\evec_2\}$$. What is the relationship between $\xvec$ and $\coords{\xvec}{\ecal}$?

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