Mathematics 581    Problem Set 8 (the last set)     Do not hand in.
Solutions will be posted Wednesday, June 5

Reminder.  Final exam in our classroom, Wednesday, June 12:  9:30 am - 11:18 am.

1. Let V be a vector space over a field F.  Suppose that  {v1, ... , vn}  spans  V.  Prove that there is a subset of  {v1, ... , vn} which is a basis for  V.

2.  Let  A  be a linear transformation of a vector space  V  over  F  into a vector space  W  over  F.

(a)  Let  z  be the zero vector in  V, and  zW  be the xero vector in  W.  Prove that
zA = zW.

(b)  Prove that for all  v  in  V,  (-v)A = -(vA).

3.  Considering the field of real numbers  R  as a vector space over its subfield of rational numbers  Q,  prove that  {1,  r}  is a linearly independent set in  R if and only if  r  is irrational.

4.  Prove that if  F  is a field, then the set of two vectors {(a, b),  (c, d)}  is linearly independent if and only if  ad - bc  is not zero.  [Caution: you may have to break up the proof into several cases.]

5.  Let  R[l]  be the vector space over the field of real numbers which consists of all polynomials with real coefficients.  Define a polynomial  p(l)  to be odd if  for every real number  a,  p(-a)= -p(a)  and define a polynomial  q(l)  to be even if for every real number  a,  q(-a)= q(a).

(a) Write down three odd polynomials of different degrees.

(b)  Write down three even polynomials of different degrees.

(c)  Prove that every polynomial is the sum of an odd polynomial and an even polynomial.

(d)  Prove that if  p(l)  is both even and odd, then  p(l) = 0.

(e)  Prove that if  s(l)  is a polynomial and  s(l) = p1(l) + q1(l) = p2(l) + q2(l),  where each  pi(l)  is odd and each  qi(l)  is even, then  p1(l) =  p2(l)  and  q1(l) = q2(l).

In vector space notation, (c)  and  (d) show that if  O  is the subspace of odd polynomials and  E  is the subspace of even polynomials, then  O + E = R[l], and the intersection of O  and  E   is {0}.