Why is every scalar product equivalent to the Standard Scalar Product on $\mathbb{R}^2$

Question

I recently encountered the following claim:

Consider $\mathbb{R}^2$ endowed with an arbitrary scalar product. By choosing an orthonormal basis we may assume that the given scalar product is the Standard Scalar Product.

I do not see why this should be the case. Could you explain?

Answer 1

After fixing an othonormal basis $\{e_1,\dots,e_n\}$, if you take two vectors $v$ and $w$, you can write them as$$v=a_1e_1+\cdots+a_ne_n\text{ and }w=b_1e_1+\cdots+b_ne_n.$$And then$$\langle v,w\rangle=\langle a_1e_1+\cdots+a_ne_n,b_1e_1+\cdots+b_ne_n\rangle=\sum_{k=1}^na_kb_k,$$as with the standard inner product and the standard basis.