Problem Set #1
Due: Monday, January 16, 2017
1. Give an example of a nonempty subset U of R2 such that U is closed under
addition and under taking additive inverses (meaning u U whenever u U ),
but U is not a subspace of R2 .
2. Suppose b R. Show that the
Problem Set #4
Due: Monday, February 6, 2017
1. Let V be a finite dimensional vector space and consider S, T, U Hom(V, V ) =
(a) Prove that ST = I if and only if T S = I.
(b) Let ST U = I. Show that T is invertible and T 1 = U S.
(c) Give an exam
1. Suppose that a0 , . . . , am are distinct elements in F and that b0 , . . . , bm are elements in F.
Prove that there exists a unique polynomial p F[t]m such that p(aj ) = bj for 0 j m.
Solution. Consider T Hom(F[t]m , Fm+1 ) defined by T p
1. Let L : C 2 ([0, 1]) C([0, 1]) be defined by Lf = f 00 .
(a) Show that L has no left inverses.
(b) Show that the operators G1 and G2 , defined as follows, are right inverses:
(G1 f )(x) =
(x t)f (t) dt ,
x(y 1) x < y
g(x, y)f (y)