16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 6.4.1

Elementary Canonical Forms

Mathematics — Linear Algebra · invariant-subspaces

Problem

Let

T

be the linear operator on

R^2

, the matrix of which in the standard ordered basis is

A = \begin{bmatrix} 1 & -1 \\ 2 & 2 \end{bmatrix}.

(a) Prove that the only subspaces of

R^2

invariant under

T

are

R^2

and the zero subspace. (b) If

U

is the linear operator on

C^2

, the matrix of which in the standard ordered basis is

A

, show that

U

has

1

-dimensional invariant subspaces.