A question in my textbook reads:
Let
$\vec{v}_1=\begin{bmatrix} -1 \\ 2 \end{bmatrix}$
and $\vec{v}_2=\left[\begin{array}{r}2\\-3\end{array}\right]$. Then $\mathcal{B}=\{\vec{v}_1,\vec{v}_2\}$ is a basis for $\mathbb{R}^2$. Please check for yourself that $\left[\begin{array}{r}x\\y\end{array}\right]=(3x+2y)\vec{v}_{1}+(2x+y)\vec{v}_{2}$. What are the coordinates of $\vec{x}=\left[\begin{array}{r}x\\y\end{array}\right]$ relative to $\mathcal{B}$?
So I have $$\mathcal{B}=\left\{\vec{v}_{1}=\left[\begin{array}{r}-1\\2\end{array}\right],\vec{v}_{2}=\left[\begin{array}{r}2\\-3\end{array}\right]\right\}$$ which forms a basis of $\mathbb{R}^{2}$, and I want to find the coordinate vector of $\left[\begin{array}{r}x\\y\end{array}\right]$ with respect to $\mathcal{B}$. Wouldn't I calculate $$\left[\begin{array}{rr}-1&2\\2&-3\end{array}\right]\left[\begin{array}{r}x\\y\end{array}\right]=\left[\begin{array}{r}-x+2y\\2x-3y\end{array}\right]?$$
Unless I am completely misunderstanding the material in this chapter, wouldn't $(3x+2y)\vec{v}_{1}+(2x+y)\vec{v}_{2}$ imply that the coordinate vector is $\left[\begin{array}{r}3x+2y\\2x+y\end{array}\right]$?
What could I be misunderstanding? Why are the answers different?
Best Answer
There are only two questions that they could be asking are this:
and
Now, if they're asking the first one (which I think they are), the answer is
$$ \begin{bmatrix} x\\y \end{bmatrix} = \begin{bmatrix} 3x+2y \\ 2x + y \end{bmatrix}_{\mathcal B} $$
If they mean the second one, the answer is
$$ \begin{bmatrix} x\\y \end{bmatrix}_{\mathcal B} = \begin{bmatrix} -x+2y \\ 2x + -3y \end{bmatrix} $$
If their answers are something other than one of these two things, then somebody made a typo, either with the question or the answer. I hope that helps.