16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 10.2.13

Bilinear Forms

Mathematics — Linear Algebra · dual-basis-bilinear-form

Problem

Let

V

be a finite-dimensional vector space and

f

a non-degenerate symmetric bilinear form on

V

. Associated with

f

is a 'natural' isomorphism of

V

onto the dual space

V^*

, this isomorphism being the transformation

L_f

of Section 10.1. Using

L_f

, show that for each basis

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

V

there exists a unique basis

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

V

such that

f(\alpha_i, \alpha_j') = \delta_{ij}

. Then show that for every vector

\alpha

V

we have

\alpha = \sum_i f(\alpha, \alpha_i') \alpha_i = \sum_i f(\alpha_i, \alpha) \alpha_i'.