Bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space. It is called bilinear because it is linear in each of its arguments. Matrix multiplication is an example.
Definition
Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function
- B : V × W → X
such that for any w in W the map
- v ↦ B(v, w)
is a linear map from V to X, and for any v in V the map
- w ↦ B(v, w)
is a linear map from W to X.
In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed. Note that if we regard the product V × W as a vector space, then B is not a linear transformation of vector spaces, except in the case of a null map, defined by B(v, w) = 0 for all (v,w) in V × W, because we would have B(v, w) = B(v, 0) + B(0, w) = 0 + 0 = 0.
If V = W and we have B(v, w) = B(w, v) for all v, w in V, then we say that B is symmetric.
The case where X is the base field F, and we have a bilinear form, is particularly useful (see for example scalar product, inner product and quadratic form).
The definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ring R. It also can be easily generalized to n-ary functions, where the proper term is multilinear.
For the case of non-commutative rings R and S, a left R-module M and a right S-module N, we can define a bilinear map as a map B : M × N → T with T an (R, S)-bimodule, and for which any n in N, m ↦ B(m, n) is an R-module homomorphism, and for any m in M, n ↦ B(m, n) is an S-module homomorphism. This is satisfies
- B(r ⋅ m, n) = r ⋅ B(m, n)
- B(m, n ⋅ s) = B(m, n) ⋅ s
for all m in M, n in N, r in R and s in S, as well as B being additive in each argument.
Properties
A first immediate consequence of the definition is that B(x,y) = 0 whenever x = 0 or y = 0. (This is seen by writing the null vector 0 as 0·0 and moving the scalar 0 "outside", in front of B, by linearity.)
The set L(V,W;X) of all bilinear maps is a linear subspace of the space (viz. vector space, module) of all maps from V×W into X.
If V, W, X are finite-dimensional, then so is L(V,W;X). For X = F, i.e. bilinear forms, the dimension of this space is dim V × dim W (while the space L(V×W;F) of linear forms is of dimension dim V + dim W). To see this, choose a basis for V and W; then each bilinear map can be uniquely represented by the matrix B(ei,fj), and vice versa. Now, if X is a space of higher dimension, we obviously have dim L(V,W;X) = dim V × dim W × dim X.
Examples
- Matrix multiplication is a bilinear map M(m,n) × M(n,p) → M(m,p).
- If a vector space V over the real numbers R carries an inner product, then the inner product is a bilinear map V × V → R.
- In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear map V × V → F.
- If V is a vector space with dual space V∗, then the application operator, b(f, v) = f(v) is a bilinear map from V∗ × V to the base field.
- Let V and W be vector spaces over the same base field F. If f is a member of V∗ and g a member of W∗, then b(v, w) = f(v)g(w) defines a bilinear map V × W → F.
- The cross product in R3 is a bilinear map R3 × R3 → R3.
- Let B : V × W → X be a bilinear map, and L : U → W be a linear map, then (v, u) ↦ B(v, Lu) is a bilinear map on V × U.
See also
External links
- Hazewinkel, Michiel, ed. (2001), "Bilinear mapping", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
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