I have been trying to find an answer to this question for some time and haven't had any luck. Let me state the question formally:
Suppose $T$ is a linear mapping $T:V \rightarrow V$, where $V$ is an $n$-dimensional vector space, $n<\infty$. Suppose $W\subset V$ is an invariant subspace of $V$, i.e., $W$ is a subspace such that $T(W) \subseteq W$, with $\dim(W) = k$. Is it always the case that we can find $k$ generalized eigenvectors of $T$, $v_1,\ldots,v_k$, such that $\text{span}(v_1,\ldots,v_k) = W$?
It seems to me that this must be true, but I've had no luck proving it on my own. I've searched around and found lots of questions/answers about the "converse" question. For example, questions about proving that a generalized eigenspace (or the direct sum of several generalized eigenspaces) is $T$-invariant. That's not what I'm after here.
Any help would be appreciated.