Thread: vector space prove!

prove wether the follow is a vector space or not.

could you please show me how to prove the following question? Intuituively, i think it is a vector space, but i don't know how to prove the 10 axioms step by step. Thank you very much.

Q1) S = {[x p(x)] | x is an element of R}, Where p(x) is any polynomial, vector addition is defined as
[x p(x)] + [y p(y)] = [x+y p(x+y)]
and scalar multiplication defined as
* [x p(x)] = [ *x p( *x)]

Just to verify, vector addition is defined as: ?

Let's go through some of the axioms and see if you can do the rest.

1.

Let's choose two arbitrary vectors, ap(a) and bp(b). Then, as it is in the form xp(x) (x = a + b) in this case.

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2. Given scalar k, , then as well.

Again, pick an arbitrary vector ap(a). Then (imagine x = ka).

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3.

Pick two arbitrary vectors ap(a) and bp(b). Then:


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4.

Pick arbitrary vector ap(a). Then, (imagine x = (c+k)a)

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5. There exists a zero vector, , such that

Again, choose arbitrary vector ap(a). Then:
(just going by definition)


etc. etc.

As you can see, verifying some of these are pretty trivial. Now, it looks like the set is indeed a vector space but I may be wrong. Go through all the axioms and see if they are all satisfied.