Let $B$ be a basis for the vector space $V$ and define $\langle,\rangle$ by $\langle u,v \rangle = [u]_B \cdot [v]_B$ for $u,v \in V$. Show that $\langle,\rangle $ is an inner product on $V$.
What I know is that it must adhere the following axioms in order to be considered an inner product:
- $\langle u,v \rangle = \langle v,u \rangle$ [Symmetry axiom]
- $\langle u + v,w \rangle = \langle u, w \rangle + \langle v, w \rangle$ [Additivity axiom]
- $\langle ku +,v\rangle = k \langle v,u \rangle$ [Homogeneity axiom]
- $\langle v,v\rangle \geq 0$ [Positivity axiom]
The abstractness of this proof is throwing me off, any help would be greatly appreciated.
Best Answer
@AidanSims's comment is awesome! However, I would like to share my step-by-step answer.
Let $B=\{v_1,v_2,\ldots,v_n\}$. Given $u,v,w\in V$, write $u=\sum_{i=1}^na_iv_i$, $v=\sum_{i=1}^nb_iv_i$, and $w=\sum_{i=1}^nc_iv_i$ for some scalars $a_i,b_i,c_i\in\mathbb{R}$.