Homework 1
Math 108A
Due Friday, October 9, 2020
Proof Problems
Your answers to the problems in this section should be proofs, unless otherwise
stated. F is a field, and V and W are vector spaces over F.
1) Are the following identities true for all sets A,B, and C? If so, prove the
identity. If not, find a counterexample. (For a counterexample, you can just draw
a Venn diagram.) Here A − B means “the set of all elements in A which are not
in B”.
a) (A−B) ∩ (C −B) = (A ∩ C)−B
b) (A−B) ∩ (C −B) = A− (B ∪ C)
c) If A ⊆ B, then A ∪ C ⊆ B ∪ C
d) If A ⊆ B, then A ∩ C ⊆ B ∩ C
e) If A ∩ C = B ∩ C, then A = B
f) If A ∪ C = B ∪ C, then A = B
g) (A ∪B) ∩ C = A ∪ (B ∩ C)
h) If A * B and B * C, then A * C
i) If A ⊆ C and B ⊆ C, then A ∪B ⊆ C
j) (A ∪B) ∩ C = (A ∩ C) ∪ (B ∩ C)
2) Let a ∈ F, and let ~v ∈ V . Prove the following identities. Keep careful track of
which axioms and results you use!
a) a ·~0 = ~0
b) (−1) · ~v = −~v
c) −(a · ~v) = (−a) · ~v.
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d) −(−~v) = ~v
e) −~0 = ~0.
3) Let S be any set, and let Fun(S, V ) the set of all functions f : S → V .
Define addition and scalar multiplication on Fun(S, V ) by
(f + g)(s) = f(s) + g(s)
(a · f)(s) = a · (f(s))
Prove that Fun(S, V ) is a vector space over F.
4) Prove that the Cartesian product V × W = {(v, w) | v ∈ V,w ∈ W} is a
vector space over F, with addition and scalar multiplication defined component-
wise.
5) There are many fields besides the ones listed in class! Consider the setQ(
√
2) =
{a+ b√2 | a, b ∈ Q}. Note that Q(√2) is a subset of R, so we can add and mul-
tiply elements of Q(
√
2). Prove that:
a)Q(
√
2) is closed under addition and multiplication; that is, for any two elements
of Q(
√
2), their sum and product both lie in Q(
√
2).
b) Prove that Q(
√
2) satisfies the field axioms. (Most, but not all, of the axioms
follow immediately from the fact that R is a field. Pay close attention to axioms
M4 and A4!)
Challenge Problem: Prove Theorem 1.3 in the lecture notes. (Hint: Use The-
orem 1.2)
Computational Problems
You don’t need to prove your answers to the following questions, but you should
still show your work.
1) List the elements of F7. Compute the additive inverse of every element and
the multiplicative inverse of each nonzero element.
2) In any field, you can add, subtract, multiply, and divide, and so you can solve
systems of linear equations using the normal methods. Solve the following linear
2
equations over the given field.
a)
{
(2 + i)x+ (3− 2i)y = i
(4 + 2i)x+ (−2− 3i)y = 1 + i ; over C.
b)
{
3x+ 2y = 5
4x+ 6y = 1
; over F7.
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