1 Linear Algebra Professor M. Zuker
Problem Set 5
Chapter 5
Note: Z1, Z2 etc. refer to my own problems, which may be very close to
what is in the text.
3. Prove or give a counterexample: if Uis a subspace of Vthat is invariant
under every operator on V, then U={0}or U=V.
4. Suppose that S, T ∈ L(V) are such that ST =T S . Prove that null(T−λI)
is invariant under Sfor every λ∈F.
Z1. Let T∈ L(R3) be defined by T(x, y, z) = (3x+2y+2z, 2x+3y+4z, −3x−3y−
4z). Let v= (1,0,0). Compute v, T v , T 2vand T3v. Show that (v, T v, T 2v) is
a basis for Vand write T3vas a linear combination of v,T v and T2v. That is,
T3v=a0v+a1T v +a2T2v. If p(x) = x3−a2x2−a1x−a0, then p(T)v= 0 and
there is no polynomial, q(x), of lower degree such that q(T)v= 0. Factor the
polynomial: p(x) = (x−λ1)(x−λ2)(x−λ3). Then (T−λ2I)(T−λ3I)vis an
eigenvector for eigenvalue λ1. Similarly, (T−λ1I)(T−λ3I)vis an eigenvector
for eigenvalue λ2and (T−λ1I)(T−λ2I)vis an eigenvector for eigenvalue λ3.
Compute these eigenvectors explicitly using the above formulas and verify
that they are indeed eigenvectors.
Z2. (Problem 8 plus extra).
1. Find all eigenvalues and eigenvectors of the backward shift operator
SB∈ L(F∞) defined by
SB(z1, z2, z3, . . . ) = (z2, z3, . . . ).
2. Find all eigenvalues and eigenvectors of T=S2
Bwhen F=R.
3. Find all eigenvalues and eigenvectors of the forward shift operator,
SF∈ L(C∞), defined by
SF(z1, z2, z3, . . . ) = (0, z1, z2, z3, . . . ).
9. Suppose T∈ L(V) and dim range T=k. Prove that Thas at most k+ 1
distinct eigenvalues.
