Disproving a claim regarding a linear transformation

Question

I received the following claim:

Let $T:V \to V$ be a linear transformation and $U$ a subspace of $V$. if $T(U) \subseteq U$, then $T(U^\bot) \subseteq U^\bot$.

I am having issues disproving the claim. I tried setting $T:R^2 \to R^2$ by $T(v)=proj_uv$ and $U=span \lbrace \begin{bmatrix}1 \\0 \\\end{bmatrix} \rbrace $

Any assistance will be welcomed.

Answer 1

The claim is false without more restrictions.

As a counterexample let $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$, such that $T(e1) = e1$ and $T(e2) = e1$. Suppose we have the usual inner product. Let $U = span<e_1>$. Then $U^{\perp} = span<e_2>$