Take any $n\times n$n matrix $\def\R{{\Bbb R}}A \in M_{n \times n}(\R)$ with real entries. Let $T_\R:\R^n \rightarrow\R^n$ and $\def\C{{\Bbb C}}T_\C:\C^n \rightarrow\C^n$ be the corresponding linear maps (defined by $T_\R(x)=Ax$ and $T_\C(v)=Av$ for $x \in\R^n$ and $v \in\C^n$). If $W \subset\C^n$ is any $T_\C$-invariant $\C$-subspace of $\C^n$, prove that $\overline{W}$ is $T_\C$-invariant, and $\operatorname{Re}(W)$ is $T_\R$-invariant.
To prove invariant, it suffices to prove that $T_\C(\bar{v}) \in \overline{W}$
let $v=x+i y$, then its complex conjugate is $\bar{v}=x-iy$
Since $W$ is invariant, we have that for any $v \in W$, we have $T_\C(v)=Av \in W$. I am not sure how to continue on this.
First write the definition of $\overline{W}$: $$\overline{W} := \{x-iy ~|~ x+iy \in W\}$$ By definition, for any $\bar{v} = x-iy \in \overline{W}$ there exists a corresponding $v=x+iy \in W$. So $Av = Ax+iAy \in W$ and by definition $A\bar{v} = Ax-iAy \in \overline{W}$. So we can conclude that $\overline{W}$ is $T_C$-invariant.
Similarly write the definition of $\operatorname{Re}W$: $$\operatorname{Re}W := \{x ~|~ x+iy \in W\}$$ By definition, for any $x \in \operatorname{Re}W$ there exists a corresponding $v=x+iy \in W$. So $Av = Ax+iAy \in W$ and by definition $Ax \in \operatorname{Re}W$. So we can conclude that $\operatorname{Re}W$ is $T_R$-invariant.