Page 275 - Advanced Linear Algebra
P. 275
Chapter 11
Metric Vector Spaces: The Theory of
Bilinear Forms
In this chapter, we study vector spaces over arbitrary fields that have a bilinear
form defined on them.
Unless otherwise mentioned, all vector spaces are assumed to be finite-
-
dimensional. The symbol denotes an arbitrary field and - denotes a finite
field of size .
Symmetric, Skew-Symmetric and Alternate Forms
We begin with the basic definition.
-
Definition Let = be a vector space over . A mapping ºÁ »¢ = d = ¦ - is
called a bilinear form if it is linear in each coordinate, that is, if
º % b &Á'» ~ º%Á'» b º&Á'»
and
º'Á % b &» ~ º'Á %» b º'Á &»
A bilinear form is
1 symmetric if
)
º%Á &» ~ º&Á %»
for all %Á & = .
)
2 skew-symmetric ( or antisymmetric ) if
º%Á &» ~ cº&Á %»
for all %Á & = .
