Thread: Inner product space

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

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