I am studying Linear Algebra and my professor proposed an exercise:
Let $\mathbb{R^n}$ a vector space, $v=(v_1,...,v_n)$ and $u=(u_1,...,u_n)$, then there is a general form for Inner Product between $u$ and $v$?
My attempt:
I know that the inner product is bilinear, comutative, $\langle u , u \rangle \geq 0 $ and $\langle u,u \rangle = 0 \iff u=0.$ So I think that the general form of Inner Product is:
$ \langle u,v \rangle = \displaystyle \sum_{j=1}^n \alpha_jv_ju_j,\hspace{0.5cm}$ where $\alpha_j \geq0, \forall j=1,...,n.$
But, how I arrive in this expression? and every Inner Product has this form?