What is a trivial linear operator?

Question

The question asks to find a non-trivial linear operator T to make a subspace T-invariant. I'm thinking of $T(x)=2x$, since $2x$ clearly stays in the subspace by scalar multiplication closure, but not sure if this is a "trivial" solution...

Answer 1

Your question asks for a nontrivial linear operator $T$ on the space of $n \times n$ such that the sets of symmetric and skew-symmetric matrices are each $T$-invariant.

In this context, I'd consider $T = \lambda I$ to be a trivial solution, because it leaves every subspace invariant, so you're not even using the stuff about symmetric and skew-symmetric matrices.

The obvious solution here would be to take something like $A \mapsto A^\top$. The symmetric and skew-symmetric matrices are clearly invariant, but most subspaces aren't.