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?