# Math Help - Inner product space

1. ## Inner product space

Show that if $V = \mathbb C^n$
, then for any inner product on V , there
is a unique complex n × n matrix A such that $\langle \vec{v},\vec{w}\rangle =\vec{v}^T A \overline{\vec{w}}$ for
all $\ \vec{v},\vec{w}\in V$.
How to show existence?

2. What happens to the usual basis vectors, i.e. $$ where $e_i$ is the vector with 1 in the ith coordinate and 0 elsewhere? Note that, assuming such a representation is always possible, $=a_{ij}$.