Kunze Linear Algebra · Problem 6.7.2

Elementary Canonical Forms

Mathematics — Methods · no-complementary-invariant-subspace

Problem

Let TT be the linear operator on R2R^2, the matrix of which in the standard ordered basis is [2102].\begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}. Let W1W_1 be the subspace of R2R^2 spanned by the vector ϵ1=(1,0)\epsilon_1 = (1, 0). (a) Prove that W1W_1 is invariant under TT. (b) Prove that there is no subspace W2W_2 which is invariant under TT and which is complementary to W1W_1: R2=W1⊕W2.R^2 = W_1 \oplus W_2. (Compare with Exercise 1 of Section 6.5.)

Browse the problem library