Finite Dimensional Vector Spaces - Paul R. Halmos

[x0104]

Hello,

I am currently working out of FDVS - Halmos, and I was wondering if a solutions manual (for the problems at the end of each section) existed?

I'd like to be able to check my work.

Thanks,

Steve

P.S Sorry if this is an inappropriate post for this section.

 

[HallsofIvy]

The easiest way to check your work, for any problem, is to see if your answer satisifies the conditions in the problem. That is typically far easier than solving the problem initially.

 

[mathwonk]

in general, if you are unable to tell whether your answer to a problem is correct, there is something you are missing in your grasp of the subject. take as a new exercise the problem of understanding what you are doing well enough to decide this for yourself, whether it is checking a numerical answer by a different way of computing it, or just understanding the reasoning of a proof.

 

[x0104]

True. However, for problems I am unable to solve and to see how my solutions compare to what the author had in mind, a solutions guide would be ideal. Does a solutions manual exist?

Thanks,

Steve

 

[mathwonk]

you keep asking the same question. the point is if you need it repeated, solutions manuals are of no benefit to anyone. the only use i make of the one in my office is as a doorstop.

 

[x0104]

Interesting. Perhaps you can help me with a problem I am having?

Is it possible to have a vector space of an infinite set with a larger cardinality then the field on which it is defined? (example: Real numbers defined over the field of rationals) I understand that the field provides scalars such that the definition of vector spaces is satisfied.

What I am having trouble with: since the scalars help define the span of the vector space, and the rationals have smaller cardinality then the reals, would you 'run out' of rationals to describe the reals with?

Sorry if this isn't exactly articulate, but this stuff is new to me.

Steve

 

[morphism]

That's a very good question. Something you might have neglected to notice is that the basis can be very large too -- uncountable, even. So it's perfectly conceivable that there are infinite vector spaces over even finite fields, and in fact this is possible.

Now, R is certainly a vector space over Q -- however, it cannot have a finite basis, because of the reasons you mentioned (R is simply too large). So, if a basis were to exist, then it would be infinite (uncountable in fact). Proving the existence of such a basis is another matter, but maybe you've seen already that every vector space has a basis.

 

[x0104]

Oh ok. That makes sense to me. Thanks.

Well I have your attention;

In general distributivity on subspaces is not true. It is easy to show that this is the case by counter-example. Halmos however then asks us to prove that:

for L,M and N that are subspaces on a vector space show that

L intersect ( M + (L intersect N)) = (L intersect M) + (L intersect N)

Because the addition operation on subspaces may yield a resulting set with more than n (n being the dimension of the whole vector space) elements, we know that if there are more than n elements, those extra elements are repeats of, or linear combinations of the other n elements in the set. So we then drop those extra (?), and this is why in general distributivity does not hold. (The only case it does being where they are all mutually exclusive?)

I think this is at the heart of the idea but for this proof I am not sure how to proceed?

 

[morphism]

How do you usually show that two sets are equal? You prove that each contains the other! Have you tried that here?