Page 107 - Linear Algebra Done Right

Chapter 5. Eigenvalues and Eigenvectors
94
Exercises
1. Suppose T ∈L(V). Prove that if U 1 ,...,U m are subspaces of V
invariant under T, then U 1 +· · ·+ U m is invariant under T.
2. Suppose T ∈L(V). Prove that the intersection of any collection
of subspaces of V invariant under T is invariant under T.
3. Prove or give a counterexample: if U is a subspace of V that is
invariant under every operator on V, then U ={0} or U = V.
4. Suppose that S, T ∈L(V) are such that ST = TS. Prove that
null(T − λI) is invariant under S for every λ ∈ F.

2
5. Define T ∈L(F ) by
T(w, z) = (z, w).
Find all eigenvalues and eigenvectors of T.

3
6. Define T ∈L(F ) by
T(z 1 ,z 2 ,z 3 ) = (2z 2 , 0, 5z 3 ).

Find all eigenvalues and eigenvectors of T.
n
7. Suppose n is a positive integer and T ∈L(F ) is defined by
T(x 1 ,...,x n ) = (x 1 + ··· + x n ,...,x 1 +· · ·+ x n );

in other words, T is the operator whose matrix (with respect to
the standard basis) consists of all 1’s. Find all eigenvalues and
eigenvectors of T.

8. Find all eigenvalues and eigenvectors of the backward shift op-
erator T ∈L(F ) defined by
∞
T(z 1 ,z 2 ,z 3 ,...) = (z 2 ,z 3 ,...).

9. Suppose T ∈L(V) and dim range T = k. Prove that T has at
most k + 1 distinct eigenvalues.

10. Suppose T ∈L(V) is invertible and λ ∈ F \{0}. Prove that λ is
an eigenvalue of T if and only if 1 λ is an eigenvalue of T −1 .