Is the following Proof Correct? In particular can the argument be stated in a better manner?
Proposition. Let $V$ an $n$ dimensional vector space and and let $T:V\to V$ be a Linear Transformation. Suppose $W$ is a $T$-invariant subspace of $V$ of dimension $k$. Show that there is a basis $\beta$ for $V$ such that $[T]_\beta$ has the form $$\begin{pmatrix} A& B\\ O&C \end{pmatrix}$$ where $A$ is a $k\times k$ matrix and $O$ is the $(n-k)\times k$ zero matrix.
Proof. Let $\beta = \{w_1,w_2,...,w_k,w_{k+1},w_{k+2},...,w_n\}$ be a basis for $V$ obtained by extending the basis $\alpha = \{w_1,w_2,...,w_k\}$ of $W$. Now consider the vectors $T(w_1),T(w_2),...,T(w_n)$. The $T$-invariance of $W$ implies that for $j = 1,2,...,k$ we have $T(w_i)\in W$ consequently $T(w_j) = \sum_{i=1}^{k}A_{ij}w_i$ and thus ${[T]_{\beta}}_{ij} = 0$ for $i = (n-k),(n-k)+1,...,n$ with these deductions it follows that $[T]_\beta$ has the required form.
$\blacksquare$