Let T : V → V be a linear operator on a finite-dimensional vector space V , such that any subspace W of V satisfies T(W) ⊆ W. (a) Show that for each non-zero vector v ∈ V , there exists a scalar av...

Let T : V → V be a linear operator on a finite-dimensional vector space
V , such that any subspace W of V satisfies T(W) ⊆ W.
(a) Show that for each non-zero vector v ∈ V , there exists a scalar av depending
on v such that T(v) = avv.
(b) Suppose that {u, v} is a linearly independent set of vectors in V . Show that
the scalars au and av in part (a) are equal. (Hint: Consider T(u + v).)