Norm of vector equals norm of it's basis representation

Question

I will try to represent my question by example.

There is a vector $a \in R^d$, basis $b$ spans $R^d$, so vector $a=\sum_{i=1}^{d}c_i b_i$.

Whether $\left \| a \right \| = \left \| c \right \|$? If yes, why?

Answer 1

Only if $b$ is an orthonormal basis. In that case we have $$ ||a|| = ||\sum_{i=1}^d c_i b_i || = \sum_{i=1}^d |c_i| ||b_i|| = \sum_{i=1}^d |c_i | = ||c||. $$ If $b$ is not orthonormal, either $||b_i|| \neq 1$ which makes the third inequality untrue (in general), or the $b_i$ are not perpendicular, which (in general) turns the second equality into an inequality using the triangle inequallity.