For a linear transformation, $B:V\rightarrow V$, on a non-standard basis, $\beta$, is the resulting solution's coordinates in $\beta$ too?
So if $B: \mathbb{R} ^2 \rightarrow \mathbb{R} ^2$, on a basis $\beta={\begin{bmatrix} 1\\2\end{bmatrix}, \begin{bmatrix} 0\\1\end{bmatrix}}$, and I transform an $[\vec{x}]_\beta = \begin{bmatrix} 2\\2\end{bmatrix} $, is the solution to $\beta [\vec{x}]_\beta$ a set of coordinates in $\beta$ or in the standard basis?
In $V:={\mathbb R}^2$ you can adopt the standard basis or some other basis $\beta=(b_1,b_2)$, $b_i\in V$. A given linear transformation $B:\>V\to V$ has a certain matrix $[B]$ with respect to the standard basis and some other matrix $[B]_\beta$ with respect to the basis $\beta$; but you have not specified either of them. Given a column vector $[x]_\beta$, to be interpreted as coordinate vector of $x\in V$ with repect to the basis $\beta$, the matrix product $[B]_\beta\>[x]_\beta$ gives you the $\beta$-coordinates of the vector $Bx\in V$.