## Halmos Sections 2,3, and 4

1.

**Problem:**

Prove that if is a vector space over the field , then for any and the following are true:

a)

b)

c)

d)

e) If the either or

f)

g)

**Proof:**

a) This follows from and the commutativity of

b) We merely note that and so

c) We merely note that and thus by cancellation

d) We see that

e) This is identical to the similar problem in the last post.

2.

**Problem: **If is a prime then is a vector space over . How many vectors are there in this vector space?

**Proof: **This is equivalent to asking how many functions are there from to which is

3.

**Problem: **Let be the set of all ordered pairs of real numbers. If and are elements of write , , and

**Proof: **It is not. Notice that and yet which contradicts the e) in the problem one.

**4.**

**Problem: **Sometimes a subset of a vector space is itself a vector space. Consider, for example, the vector space and the subsets of consisting of those vectors such that

a) is real

b)

c) Either or

d)

e)

**Proof:**

a) This clearly isn’t (remembering that we’re considering as being a vector space over ) since but

b) It suffices to show that , , and since all the attributes of a vector space (concerning the addition and scalar multiplication) are inherited. But, all three are glaringly obvious. So yes, this is a subspace.

c) No, note that but

d) Clearly . Also, if we have that since . Lastly, if we see that since .

e) No, consider that but

5.

**Problem: **Consider the vector space (the set of all complex coefficiented polynomials) and the subsets consisting those vectors for which

a)

b)

c)

d)

Which of them are vector spaces?

**Proof:**

a) This is not since the zero function isn’t in it.

b) This is.

c) This isn’t since but

d) This is. (maybe, I got a little lazy)

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