A question about vector space, sub-spaces and linear transformation

Question

I have a homework question that I don't know how to start:

let $V$ be a vector space under field $F$, $\dim(V) = n$.

let $T: V \to V$ be a linear transformation.

prove if $U \subseteq V$, and $\dim(U) = k$, such that for every $u \in U$, $T(u) \in U$, then there exists a basis $B$ of $V$ such that:

$[T]^B_B =$ $ \begin{bmatrix} A_1 & A_3 \\ 0 & A_2 \\ \end{bmatrix} $

So that $A_3 ∈ M_{k×(n−k)}(F)$, $A_2 \in M_{(n−k)×(n−k)}(F)$ and $A_1 \in M_{k×k}(F)$

Thank you for your help!

Answer 1

Hint: Take a basis $u_1,\dots,u_k$ of $U$ and complete it to a basis $u_1,\dots,u_k,u_{k+1},\dots,u_n$ of $V$.

Answer 2

Hint: try to construct the basis $B$. Start with a basis of $U$ ($k$ vectors) and complete it to a basis in $V$ by $n-k$ linearly independent vectors. How does the matrix of $T$ look like in this basis?

Answer 3

First you argue that there is a basis of $V$ such that any element of $U$ can be written as $(u_1,\dots,u_k,0,\dots,0)\in V$.

Using this and that by assumption $T(U)\subset U$ we deduce that the matrix representation of $T$ in this basis will be: $$ \begin{bmatrix} A & B \\ C & D \\ \end{bmatrix} $$ where $A$ is a $k\times k$ matrix and $C=0$.