MATH 700 Quiz #2 Name:
Fall, 2008
1. Let nbe a positve natural number, Abe an n×nmatrix over a field F, and
T∈L(Fn) . In each case determine if Aor T, respectively, is invertible, not
invertible, or there is not sufficient information to decide. Justify your answer.
a. Tk= 0 for some k≥2 .
b. AB = 0 for some nonzero n×pmatrix Bwith p≥1 .
c. Ais similar to an invertible n×nmatrix B.
d. nullity(T)>rank(T) .
2. Suppose Vis an n-dimensional vector space, n > 0 , and T∈L(V) . Let v
be a non-zero vector in V. Explain why α=hv, T v, T 2v, . . . , T nvimust be
dependent, and why span(α) must be T-invariant.
