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 ) =
End(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
Solutions #5
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
Solutions #4
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:
Z x
(G1 f )(x) =
(x t)f (t) dt ,
0
Z 1
x(y 1) x < y
g(x, y)f (y)
Problem Set #9
Due: Monday, March 20, 2017
1. Let V be a complex inner-product space. Suppose S End(V ) is a normal
operator satisfying S 9 = S 8 . Prove that S is self-adjoint and S 2 = S.
2. Suppose that T End(V ) is self-adjoint, K and > 0. Prove that
Problem Set #8
Due: Monday, March 13, 2017
1. Find a polynomial p R[x]2 such that p(0) = 0 and
Z 1
|3 + 4x p(x)|2 dx
0
is as small as possible.
2. Find g R[t]2 such that
R1
0
f (t) cos(t) dt =
R1
0
f (t)g(t) dt for all f R[t]2 .
3. Let V be the R-subspace
Solutions #7
1. Find a polynomial p R[x]2 such that p(0) = 0 and
Z
1
|3 + 4x p(x)|2 dx
0
is as small as possible.
R1
Solution. Consider the inner product space V := R[x]2 where hf, gi = 0 f (x)g(x) dx.
Let U := cfw_f V : f (0) = 0. If f, g U and r R, then
QUEENS UNIVERSITY
DEPARTMENT OF MATHEMATICS AND STATISTICS
MATH 212
MIDTERM EXAMINATION
VERSION 1
15 FEBRUARY 2017
PROFESSOR: MARIA AVDEEVA
Name:
Student Number:
This examination is two hours in length.
Calculators, data sheets or other aids are not per
QUEENS UNIVERSITY
DEPARTMENT OF MATHEMATICS AND STATISTICS
MATH 212
MIDTERM EXAMINATION
VERSION 2,3,4
PROFESSOR: MARIA AVDEEVA
Name:
Student Number:
This examination is two hours in length.
Calculators, data sheets or other aids are not permitted.
Each
Solutions #7
1. Suppose an orthonormal basis (e1 , . . . em ) and another list of vectors (w1 , . . . wm ) in
V satisfy
1
kej wj k < ,
m
for all j = 1 . . . m and some norm k k on V . Prove that (w1 , . . . wm ) is a basis of V .
Solution. It is enough to