$\begingroup$We know that if $v=a_1 e_1+\cdots+a_n e_n$ where $(e_{i})$ is the standard basis on $\mathbb{R}^{n}$ that the Euclidean norm is given by
$$\|v\|^2 = a_1^2 +\cdots+a_n^2.$$
Suppose that $v$ were written in component form with respect to some arbitrary basis $(B_i)$.
Is there a formula for the norm of $v$ with respect to the basis $(B_i)$? I'm certain there is, I just can't seem to find it.
$\endgroup$1 Answer
$\begingroup$For $$v=a_1 v_1+\cdots+a_n v_n$$ we get $$||v||^2 = (a_1 v_1+\cdots+a_n v_n).(a_1 v_1+\cdots+a_n v_n)=$$
$$ a_1^2 ||v_1||^2 + 2a_1a_2v_1.v_2 +....+a_n^2 ||v_n||^2$$
As you see the choice of standard basis is a smart choice.
$\endgroup$
