Kunze Linear Algebra · Problem 7.2.5

Rational and Jordan Forms

Mathematics — Methods · t-invariant-direct-sum

Problem

Let

T

be a linear operator on the vector space

V

over the field

F

. If

f

is a polynomial over

F

and

\alpha

is in

V

, let

f\alpha = f(T)\alpha

. If

V_1, \ldots, V_k

are

T

-invariant subspaces and

V = V_1 \oplus \cdots \oplus V_k

, show that

fV = fV_1 \oplus \cdots \oplus fV_k.