Given a $\mathbb{C}$-vectorspace $V=\begin{bmatrix}a&b\\-b&a\end{bmatrix}$ for $a,b∈ \mathbb{C}$ and a basis of $V$ by $B=\begin{bmatrix}1&1\\-1&1\end{bmatrix},\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ and $C=\begin{bmatrix}-1&1\\-1&-1\end{bmatrix},\begin{bmatrix}1&0\\0&1\end{bmatrix}$, set $T : V → V$ to be the linear transformation such that $$_C[T]_B = \begin{bmatrix}1+i&i\\2+i&1+i\end{bmatrix}.$$
From this, we deduce that $$T\begin{bmatrix}1&1\\-1&1\end{bmatrix}=(1+i)\begin{bmatrix}-1&1\\-1&-1\end{bmatrix}+(2+i)\begin{bmatrix}1&0\\0&1\end{bmatrix}=\begin{bmatrix}1&1+i\\-1-i&1\end{bmatrix}$$
and
$$T\begin{bmatrix}0&1\\-1&0\end{bmatrix}=(i)\begin{bmatrix}-1&1\\-1&-1\end{bmatrix}+(1+i)\begin{bmatrix}1&0\\0&1\end{bmatrix} =\begin{bmatrix}1&i\\-i&1\end{bmatrix}.$$
From the second part, we get that $A$, which is a matrix corresponding to $T$ is $\begin{bmatrix}i&-1\\1&i\end{bmatrix}$.
However, if I substitute this into the first part, it won't work:
$$\begin{bmatrix}i&-1\\1&i\end{bmatrix}\begin{bmatrix}1&1\\-1&1\end{bmatrix}≠\begin{bmatrix}1&1+i\\-1-i&1\end{bmatrix}.$$
Where have I gone wrong?
It’s unclear from your question what exactly it is that you’re meant to do in this exercise. However, when checking your work at the end it appears that you’ve made the error of confusing elements of $V$ with their coordinates relative to some basis. The former are $2\times2$ matrices with complex entries, but the latter are elements of $\mathbb C^2$, i.e., two-dimensional complex vectors. To put it another way, denoting the standard basis by $E$, you’ve computed $_E[T]_E v$ instead of $_E[T]_E[v]_E$. If $V$ had been defined as, say, the set of complex matrices of the form $\small{\begin{bmatrix}a&b&b\\-b&a&b\\-b&-b&a\end{bmatrix}}$ instead, there would’ve been no possibility of making this mistake. This space is also two-dimensional, so the matrix of any $T:V\to V$ is $2\times2$, hence the product $_E[T]_E v$ is undefined.