Proving inner product conditions.

Question

I'm new to the concept of inner product space and was wondering if anyone cold help me here.

I believe that the inner product space is the multiplication of each corresponding entry of the two vectors and then the addition of all of these values.

So I believe to show part (a) ii., I have to say that for any length of $\vec v$ you will always have $0* \vec v_{n} =0$ and that $0+0+0+0...$ n times will always result in 0.

However for part (a) i., I don't know how to approach this at all.

Am I headed on the right track or is my thinking incorrect?

Any help would be appreciated.

Answer 1

for part (a) i., I don't know how to approach this at all

For (a) i., we have $\langle \vec u,\vec v\rangle=\vec u^T\vec v$, so $\langle M\vec u,\vec v\rangle=(M\vec u)^T\vec v.$

By a property of transpose, $(M\vec u)^T=\vec u^TM^T$.

Therefore $\langle M\vec u,\vec v\rangle= \vec u^TM^T\vec v=\langle \vec u,M^T\vec v\rangle.$