Can someone help me with this problem?
Let $V$ be an $n$-dimensional vector space, and let $T: V \rightarrow V$ be a linear transformation. Suppose that $W$ is a $T$-invariant subspace of $V$ having 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.
Attempt at a solution: Let $(v_1, v_2, \ldots, v_k)$ be a basis for $W$ and extend it to a basis $\beta = (v_1, v_2, \ldots, v_n)$ for $V$. Since $W$ is $T$-invariant, we have $\forall v_j \in W$, $T(v_j) \in W$ for $j= 1, 2, \ldots, k$. Hence $T(v_j) = \sum_{i=1}^{k} a_{ij} v_i$.
Don't know where to go from here. How to find A, B, C and $O$?
Best Answer
Let $T: V \rightarrow V$ be a linear transformation and $\beta = \{ v_1, \dots, v_n \}$ be a basis constructed as to place $\{ v_1, \dots , v_k \}$ in $W$. Define $$ [T]_{\beta} = [[T(v_1)]_{\beta}|\cdots | [T(v_k)]_{\beta}|[T(v_{k+1})]_{\beta}|\cdots | [T(v_n)]_{\beta}] $$ Here $[c_1v_1+ \cdots +c_nv_n]_{\beta} = [c_1, \cdots , c_n]^T$. This is the $\beta$-coordinate map. By the $T$-invariance of $W$ we have the existence of $a_{ij}$ for $1 \leq i,j \leq k$ such that $$ T(v_j) = \sum_{i=1}^k a_{ij}v_i $$ now, just to emphasize what's happening, I'll focus on $j=1$, $$ T(v_1) = \sum_{i=1}^k a_{i1}v_i = a_{11}v_1+ \cdots +a_{k1}v_k+0v_{k+1}+ \cdots +0v_n $$ of course we can easily read off the coordinates of $T(v_1)$ given the expression above: $$ [T(v_1)]_{\beta} = [a_{11}, \dots, a_{k1}, 0, \dots , 0]^T$$ Likewise, for $j=2, \dots k$ we find $k$-possibly nonzero coefficients in the image paired with $n-k$ necessary $0$'s. In contrast, we get to say nothing about $T(v_{k+1})$ to $T(v_n)$. So, what does that mean?