Be careful! If n= 1, these are NOT true!

No, you don't. You don't need to show that the zero vector is in the subset because you have already shown if v is any vector in the set and a any number, av is in the set. Take a= 0. (You DO have to show that the set is non-empty. That's why you need n>1.) You don't need to show, for example, that addition is associative, commutative, etc. because you are given that V is a vector space so those must be true.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 .

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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.

Y are given the single vector v. You can always choose a basis containing that vector. Can you use the Gram-Schmidt process to construct an orthonormal basis from that? Once again, for this to be true you must have n> 1.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).