Abstract Linear Algebra See image Suppose V is finitedimensi

Abstract Linear Algebra. See image

Suppose V is finite-dimensional, T epsilon Pound (V), and U is a subspace of V. Prove that U and U are both invariant under T if and only if PuT = TP_U.

Solution

Suppose U and U are invariant under T.

Write v V as v = u + w with u U,

w U . Then P_U T v = P_U (T u + T w) and T u U,

T w W implies P_U (T u + T w) = T u.

Meanwhile, T P_U (u + w) = T u

so T P_U v = P_U T v for every v V

. Now suppose that T P_U = P_U T.

Take u U. Then T u = T P_U u = P_U T u

which is an element of U since range P_U = U.

Thus T u is contained in U. Now take w U .

We have P_U T w = T P_U w = T0 = 0.

It follows that when we write T w = u + w with u U, w U we have u = 0.

Thus T w U so U is also T-invariant.