simpletex Assignment 7
Assignment 7
(1) Let V be a 3-dimensional vector space, and let S, T ∈ L(V ). Suppose
that S and T each have 1, 3, and 5 as eigenvalues. Show that there
exists an invertible operator R ∈ L(V ) such that S = R−1TR.
(2) Suppose that V is a finite-dimensional vector space, and let T ∈
L(V ).
(a) Show that if NullT ∩RangeT = {0}, then V = NullT⊕RangeT .
(b) Show that if T 2 = T , then V = NullT ⊕ RangeT .
(c) Give an explicit example of a non-zero operator T ∈ L(R3)
such that T 2 = T . If you define you operator via a matrix, you
should also explain in words what it does to an arbitrary vector.
(d) Show that V = NullT ⊕ RangeT does not imply T 2 = T by
showing that for any diagonalisable operator S, V = NullS ⊕
RangeS.
(3) Let T ∈ L(Fn) for n < ∞. Show that if every (n − 1)-dimensional subspace of Fn is T -invariant, then T = cI for some c ∈ F. (4) Let F denote the vectors space of “Fibonacci-type” sequences of real numbers, F = {(a0, a1, a2, . . . ) | ai ∈ R and an+2 = an + an+1}. This is a 2-dimensional real vector space. (If you haven’t thought about this set before, you should convince yourself of this first.) Let S ∈ L(F) be the shift operator defined by S(a0, a1, a2 . . . ) = (a1, a2, . . . ). Show that an eigenvector for S must be a geometric sequence and find the eigenvalues for S. 1