Bilinear Forms Definition 9. A bilinear form on a pair of vector spaces V and W
ID: 1948334 • Letter: B
Question
Bilinear FormsDefinition 9. A bilinear form on a pair of vector spaces V and W over R is a function B
which assigns to each ordered pair (v,w) with v ? V and w ? W, a uniquely determined
element of R, denoted by B(v, w), such that the following conditions are satisfied:
1. B(v1 +v2,w)=B(v1,w)+B(v2,w) 2. B(v,w1 +w2)=B(v, w1)+B(v, w2) 3. B(?v, w) = B(v, ?w) =
?B(v, w) for all v, v1, v2 ? V, w, w1, w2 ?W and??R.
Problem 4.1. 1. Consider R3 with the standard inner product given by <v, w>= v1*w1+
v2*w2 +v3*w3 where v = (v1, v2, v3) and w = (w1, w2, w3). Show that B(v,w) :=< v,w > is a
bilinear form. Generalize this result to the standard inner product on Rn, that is,
first, define the standard inner product on Rn, then show that it is a bilinear form.
Explanation / Answer
The definition of a bilinear form can easily be extended to include modules over a commutative ring, with linear maps replaced by module homomorphisms. When F is the field of complex numbers C, one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument.
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