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**I-Think** Let V be a n-dimensional vector space, and let $\displaystyle T $ be a linear operator on $\displaystyle V$

Suppose that $\displaystyle W$ is a $\displaystyle T$-invariant subspace of $\displaystyle V$, $\displaystyle dim(W)=k$

Show that there exists a basis $\displaystyle \alpha $ for $\displaystyle V$ such that $\displaystyle [T]_{\alpha}$ has the form

$\displaystyle \[

\left( {\begin{array}{cc}

A & B \\

O & C \\

\end{array} } \right)

\]$

where $\displaystyle A$ is a $\displaystyle k*k$ matrix and $\displaystyle O$ is the $\displaystyle (n-k)*k$ zero matrix

Request

The question itself confuses me. I am not sure what the question wants

If $\displaystyle \alpha$ is a basis for $\displaystyle V$, then $\displaystyle \alpha$ has n vectors and

$\displaystyle [T]_{\alpha}$ is a $\displaystyle n*n$ matrix

So what is this question asking us to reduce the matrix to a $\displaystyle 2*2$ matrix with the entries themselves being matrices?