Linear Algebra

V is a 2 dimensional vector space with basis {a,b}. Find all inner products, (-,-) which satisfy (a,a)=(b,b)=1

I started by considering (u,v) and then writing u and v in terms of the basis and using the rules for inner produts to expand it as much as possible but I end up with (u,v)=c+d(a,b) where c and d are scalars determined u and v. Is this the right way to go and if so how can I get rid of the (a,b) term?

(u,v)=c+d(a,b) where c and d are scalars determined u and v.
Is this the right way to go and if so how can I get rid of the (a,b) term?

Why would you want to get rid of the $\langle a, b \rangle$? A hint is that you get an inner product for every scalar of the field.
Also, you have assumed that $\langle a, b \rangle = \langle b, a \rangle$, but you have no been told you have a real vector space.

But I don't want to have the inner product defined as a function of itself. Also we only do inner products over real v spaces at the moment.

Follow my suggestion above. You'll be able to find a restriction on the value of $\langle a,b\rangle$

I get (a,b)>0, is this all I can say?

I get (a,b)>0, is this all I can say?

I get (a,b)>0, is this all I can say?

OK, I think I finally figured this out. Hint: Cauchy Schwarz inequality.