Bilinear Forms Definition 9. A bilinear form on a pair of vector spaces V and W
ID: 1947888 • Letter: B
Question
Bilinear Forms
Definition 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) forallv,v1,v2 ?V,w,w1,w2 ?W and ??R.
Problem 4.1.
1. Consider R^3 with the standard inner product given by < v, w >= v1w1+
v2w2 +v3w3 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.
2. Let V be a vector space over R with dual vector space V?. Define B(v,f) = f(v) for v?V
and f?V?.Prove that B is a bilinear form.
Explanation / Answer
B is a bilinear form.
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