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Mathematics 108A
Homework # 1
Due: October 1, 2009
Show that Z2 = {0,1} is a field. Is Z3 = {0,1,2} a field? Z4 = {0,1,2,3}?
Let Q be the rational numbers and Q[√3] = {p + q √3 | p, q  Q}. Show that
Q[√3] is a vector space over Q.
Let X denote the collection fifth degree polynomials p(x) (in variable x with real
coefficients) with roots at x = , i.e. p() = 0. Is X a subspace of the vector
space of fifth degree polynomials with real coefficients. If yes, explain why. If
no, explain why not?
Let V be the first quadrant in the xy-plane: that is, let
 x |
V =    | x  0, y 0 
  y |
(i) If u and v are in V, is u + v in V? Why or why not?
(ii) If u is in V and c is in R, is cu in V? Why or why not?
(iii) Is V a real vector space? Explain.
Suppose that F is a field.
(i) Prove that –(-r) = r, for any rF.
(ii) Prove that (-1)(-1)=1.
(ii) Prove that if r0, (r--1)-1=r, where r—1is the multiplicative inverse of r.
Suppose that W is a vector space and wW. Prove that –(-w) = w.
Give an example of a nonempty subset U of R2 that is closed under scalar
multiplication but is not a subspace of R2.
Prove that the intersection of any collection of subspaces of a vector space is a
vector space.
Prove that the union to two subspaces of a vector space is a subspace if and only if
one of them is contained in the other.
Prove or give a counterexample: If U1, U2, and W are subspaces of a vector space
V such that U1+W = U2+W, then U1 = U2 .
Prove or give a counterexample: If U1, U2, and W are subspaces of a vector space
V such that U1W = U2W, then U1 = U2 .