Answer to Question #159354 in Linear Algebra for Abhipsita Das

Question

Let V be a finite dimensional inner product space over the field C of complex

numbers. Suppose B = {x1, x2, . . . , xn} is an orthonormal basis of V .

Prove that for all x, y ∈ V ,

(x|y) = Xn

i=1

(x|xi)(y|xi).

Answer 1

If $B=\{u_1,u_2,...,u_n\}$ is a basis of an n -dimensional inner product space $V$ , then

$x=x_1u_1+x_2u_2+,...+x_nu_n$

$x=y_1u_1+y_2u_2+,...+y_nu_n$

So:

$\langle x,y \rangle=\langle \displaystyle \sum _{i=1}^nx_iu_i,\displaystyle \sum _{j=1}^ny_ju_j \rangle$