16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 10.2.16

Bilinear Forms

Mathematics — Linear Algebra · dual-basis-columns

Problem

Let

F

be a field and

V

the space of

n \times 1

matrices over

F

. Suppose

A

is an invertible, symmetric

n \times n

matrix over

F

and that

f

is the bilinear form on

V

defined by

f(X, Y) = X^t A Y

. Let

P

be an invertible

n \times n

matrix over

F

and

\mathfrak{B}

the basis for

V

consisting of the columns of

P

. Show that the basis

\mathfrak{B}'

of Exercise 13 consists of the columns of the matrix

A^{-1}(P^t)^{-1}