Just want some feedback on my work below, I feel confident on $1,2,4$ but I'm not sure I did $3,5,6$ correctly.
In class we discussed how we could think of the modes of transportation in Gauss’ Cabin as a basis for $\mathbb{R}^2$. The travel of the hoverboard was given by $\begin{bmatrix}3 \\ 1 \end{bmatrix}$ and that of the magic carpet by $\begin{bmatrix}1 \\ 2 \end{bmatrix}$.
$1)$ Give the coordinates of Gauss’ Cabin (107 miles East and 64 miles North) using the standard basis $\varepsilon = \begin{bmatrix}1 \\ 0\end{bmatrix} , \begin{bmatrix}0 \\ 1\end{bmatrix}$
Any coordinate in the standard basis is itself so $$\begin{bmatrix}107 \\ 64 \end{bmatrix}_{\varepsilon}= \begin{bmatrix}107 \\ 64 \end{bmatrix}$$
$2)$Give the coordinates of Gauss’ Cabin using the basis based on the modes of transportation.
$$\begin{bmatrix}107 \\ 64 \end{bmatrix}_{\beta}$$ $$x*\begin{bmatrix}3 \\1 \end{bmatrix} + y*\begin{bmatrix}1 \\ 2 \end{bmatrix}= \begin{bmatrix}107 \\ 64 \end{bmatrix}$$ $$3x+y =107$$ $$x+2y=64$$ $$\begin{bmatrix}x \\ y \end{bmatrix}=\begin{bmatrix}30 \\ 17 \end{bmatrix}$$
$3)$ Suppose Uncle Cramer’s house w is located at $[w]_{\varepsilon} = \begin{bmatrix} 25 \\ 71\end{bmatrix}$. Describe this location as a vector in the ”modes of transportation” basis B.
$$\begin{bmatrix}25 \\ 71 \end{bmatrix}_{\beta}$$ $$x*\begin{bmatrix}3 \\1 \end{bmatrix} + y*\begin{bmatrix}1 \\ 2 \end{bmatrix}= \begin{bmatrix}25 \\ 71 \end{bmatrix}$$ $$3x+y =25$$ $$x+2y=71$$ $$\begin{bmatrix}x \\ y \end{bmatrix}=\begin{bmatrix}-\frac{21}{5} \\ \frac{188}{5} \end{bmatrix}$$
$4)$ Suppose you visit a museum v located at $[v]_{\beta} = \begin{bmatrix}8\\3 \end{bmatrix}$. Describe the location of the museum as a vector in the standard coordinate system $\varepsilon$.
$$x*\begin{bmatrix}1 \\ 0\end{bmatrix} + y*\begin{bmatrix}0 \\ 1\end{bmatrix}= \begin{bmatrix}8 \\ 3\end{bmatrix}$$ $$\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}8 \\ 3\end{bmatrix}$$
$5)$ Express each of the following in terms of the "modes of transportation" basis $\beta$: $[u_1]_{\varepsilon}= \begin{bmatrix}7 \\ -1\end{bmatrix}$ and $[u_2]_{\varepsilon}= \begin{bmatrix}-6 \\ 3\end{bmatrix}$
$$x*\begin{bmatrix}3 \\ 1\end{bmatrix} + y*\begin{bmatrix}1 \\ 2\end{bmatrix}= \begin{bmatrix}7 \\ -1\end{bmatrix}$$ $$\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}3 \\ 2\end{bmatrix}$$
$$x*\begin{bmatrix}3 \\ 1\end{bmatrix} + y*\begin{bmatrix}1 \\ 2\end{bmatrix}= \begin{bmatrix}-6 \\ 3\end{bmatrix}$$ $$\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}-3 \\ 3\end{bmatrix}$$
$6)$ Express each of the following in terms of the standard basis $\varepsilon$ : $$[z_1]_{\beta} = \begin{bmatrix} 7 \\−1 \end{bmatrix} and [z_2]_{\beta} =\begin{bmatrix}−6\\ 3 \end{bmatrix}$$
$$x*\begin{bmatrix}1 \\ 0\end{bmatrix} + y*\begin{bmatrix}0 \\ 1\end{bmatrix}= \begin{bmatrix}7 \\ -1\end{bmatrix}$$ $$\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}7 \\ 1\end{bmatrix}$$
$$x*\begin{bmatrix}1 \\ 0\end{bmatrix} + y*\begin{bmatrix}0 \\ 1\end{bmatrix}= \begin{bmatrix}-6 \\ 3\end{bmatrix}$$ $$\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}-6 \\ 3\end{bmatrix}$$