Let $V$ be a complex inner product space. Suppose $V$ is the orthogonal direct sum $V = U\oplus W$ of two of vector subspaces $U$ and $W$. Suppose further that $\text{dim}U = \text{dim}V = n$, $\{e_{1},...,e_{n}\}$ is a basis for $U$ and $\{f_{1},...,f_{n}\}$ is a basis for $W$.
Note that $V$ is isomorphic to $\mathbb{C}^{2n}$. If $v = u + w$, $u \in U$ and $w \in W$, then write: $$v = \sum_{i=1}^{n}(\alpha_{i}e_{i}+\beta_{i}f_{i}),$$ with $\alpha_{i},\beta_{i}\in \mathbb{C}$. Then one can send $v$ to a vector: $$v \mapsto \begin{pmatrix} \alpha_{1} \\ \beta_{1} \\ \alpha_{2}\\ \beta_{2} \\ \vdots \\ \alpha_{n} \\ \beta_{n} \end{pmatrix} \tag{1}\label{1}$$ that is, I am piling up the vectors $\binom{\alpha_{i}}{\beta_{i}}$, $i=1,...,n$.
Let us write: $$v = \begin{pmatrix} u \\ v \end{pmatrix} $$ as a column vector. Using this representation, let $A: W \to U$ be the following operator: $$A = \begin{pmatrix} 0 & I \\ 0 & 0 ,\end{pmatrix} $$ where $I$ is the identity map in $V$.
My question is: if $v$ is mapped to the piled vector as in (\ref{1}), then it seems that $A$ is mapped to: $$A = \bigoplus_{i=1}^{n}\begin{pmatrix} 0 & 1 \\ 0 & 0 .\end{pmatrix} $$ Is this true?