# Invariant subspaces under linear transformation and their complexification

The problem $$77.6$$ from finite-dimensional vector spaces:

If $$A$$ is a linear transformation on a real vector space $$V$$ and if a subspace $$M$$ of complexification $$V ^ +$$ is unchangeable under $$A ^ +$$, then $$M ^ perp cap V$$ is unchangeable under $$A$$,

The problem says that $$M ^ perp cap V$$ is unchangeable, but my solution proves it $$M cap V$$ is.

To let $$x in M ​​ cap V$$, then $$Ax = Ax + i cdot A0 = A ^ + (x + i cdot0) in M ​​$$, by the fact that $$x + i cdot0 in M ​​$$ and $$M$$ is unchangeable under $$A$$,

Is my proof correct?