Kunze Linear Algebra · Problem 10.1.3

Bilinear Forms

Mathematics — Methods · matrix-of-bilinear-form-trace

Problem

Let

V

be the space of all

2 \times 3

matrices over

R

, and let

f

be the bilinear form on

V

defined by

f(X, Y) = \text{trace}\,(X^t A Y)

, where

A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}.

Find the matrix of

f

in the ordered basis

\{E^{11}, E^{12}, E^{13}, E^{21}, E^{22}, E^{23}\}

where

E^{ij}

is the matrix whose only non-zero entry is a

1

in row

i

and column

j