Kunze Linear Algebra · Problem 8.2.14

Inner Product Spaces

Mathematics — Methods · orthonormal-bases-linear-operators

Problem

Let

V

be a finite-dimensional inner product space, and let

\mathfrak{B} = \{\alpha_1, \ldots, \alpha_n\}

be an orthonormal basis for

V

. Let

T

be a linear operator on

V

and

A

the matrix of

T

in the ordered basis

\mathfrak{B}

. Prove that

A_{ij} = (T\alpha_j|\alpha_i).