In the above image, I am having trouble understanding the solution in terms of notation. For example, take the first portion of the solution, where the intermediate step contains
$[(2, -2)]_C$
I understand in the previous matrix, $a = 1, b = 0, c = -1, d = 1$. Given the transformation definition,
$a + d = 1 + 1 = 2$
and
$-b +3c + d = 0 - 3 + 1 = -2$
Which gives $[(2, -2)]_C$.
But how is the column vector containing 2 and 0 obtained? I can't seem to figure out the operation that gives this result. An explanation for the first row of the solution would be satisfactory, since all the remaining rows follow the same general principle. For self-study- thank you!
$\begin{bmatrix} 2 \\ 0 \end{bmatrix}$ are the coefficients of the vectors $(1,-1)$ and $(1,1)$ that are needed to add up to $(2,-2)$ (i.e. to express $(2,-2)$ as a linear combination of the basis vectors $(1,-1)$ and $(1,1)$): namely, $$ (2,2) = 2(1,-1) + 0(1,1) $$ (this is perhaps clearer for the last line, where $(0,-1) = \frac{1}{2}(1,-1)-\frac{1}{2}(1,1)$, so the column vector is $\begin{bmatrix} 1/2 \\ -1/2 \end{bmatrix}$).