16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 7.2.3

The Rational and Jordan Forms

Mathematics — Linear Algebra · t-admissible-subspaces

Problem

Let

T

be the linear operator on

R^3

which is represented in the standard ordered basis by the matrix

\begin{bmatrix} 2 & 0 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}.

Let

W

be the null space of

T - 2I

. Prove that

W

has no complementary

T

-invariant subspace. (Hint: Let

\beta = \epsilon_1

and observe that

(T - 2I)\beta

is in

W

. Prove there is no

\alpha

W

with

(T - 2I)\beta = (T - 2I)\alpha