Proofs of vector space, inner product

Hi MHF,

I don't have a concrete idea about how do perform these 2 proofs.

Let be a vector space having an inner product and whose dimension is n. Let be a non zero vector and .

a)Prove that is a subspace of .

b)Prove that .

My attempt : for a)

First I notice that . Now I must show that the addition of 2 vectors in remains in and the multiplication of a vector by a scalar remains in . I understand it's obvious because multiplying a vector which is orthogonal to will give another vector orthogonal to .

Let be the inner product function. We have the property : . And I see that .

Now I must show that .

.

Wow, it seems I've show part a)! Eventually I must show other properties of , like that the zero vector is in... and so on.

I'll try part b).

I'm unable. I wish . It would be easy, because as , and and I think that , so it would have been done.

Did I do well part a)? How can I show part b)? (a tip would be welcome).