Re: What's a bilinear form?

On Tue, 08 May 2007 01:34:10 -0400, Hatto von Aquitanien
<abbot@xxxxxxxxxxxxxx> fed this fish to the penguins:

[snip]

What is there so problematic about this notion?
Bilinear< quite intuitively refers to the fact that one has
two variables in which the map is linear.
Form< refers to the fact that the values of the map lie in
the ground field.
So a bilinear form is a map

b: V x W --> K

That is what I thought, but I wanted to be sure.

where V,W are vector spaces over the field K, that
is linear in both variables.
Straight, simple and nothing to agonize about.

H

The question arrises whether the mapping is dependent upon the location
(event) where it is to be applied.

There is no such question, the above definition is strictly correct.
What you *may* want (I am guessing here) is a different, more complex
object. In physics, a metric tensor, as the word is used in gravity
for example, is a section of the second-order cotangent bundle
(T^*)^2. In other words it assigns to each point m in the manifold M
(e.g. spacetime) a bilinear form

T_m x T_m -> K

where T_m is the tangent space at m. If you choose local coordinates
around m, than this translates into a function into the linear space
of bilinear forms.

Hope it helps, best regards,
G. Rodrigues
.

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