1.2.21 Bilinear Map Definition

Bilinear Map Definition is the characterization of a bilinear map as a function of two vector arguments, drawn possibly from different vector spaces defined over a common scalar field, that is linear in each argument separately when the other argument is held fixed. It supplies the first and most fundamental case of a multilinear map, and it is the direct conceptual precursor to the tensor product construction, since every bilinear map on a pair of vector spaces corresponds, through the tensor product's universal property, to a unique linear map on their tensor product.

The Defining Condition

A function taking one vector from a first vector space and one vector from a second vector space, and producing a result in some third vector space, is called bilinear if it is linear in the first argument whenever the second argument is fixed, and linear in the second argument whenever the first argument is fixed. Linearity in each argument means the function respects both addition and scalar multiplication in that argument, exactly as required for an ordinary linear map, but this requirement is imposed separately and independently for each of the two arguments.

The expression above states linearity of a bilinear map in its first argument, with the second argument held fixed; an analogous condition, linearity in the second argument with the first held fixed, completes the definition.

Bilinearity Is Not the Same as Linearity on the Product Space

A common point of confusion is the distinction between a bilinear map and a linear map defined on the direct sum or product of the two vector spaces. A bilinear map is not, in general, linear when its two arguments are combined and treated as a single vector in a product space, because bilinearity permits multiplicative interaction between the two arguments, which an overall linear map on the product space could not produce. This distinction is precisely what motivates the construction of a new space, the tensor product, on which the corresponding operation does become an ordinary linear map, satisfying the universal property that links bilinear maps on two spaces to linear maps on their tensor product.

Common Examples of Bilinear Maps

The dot product on a real coordinate vector space is a bilinear map, taking two vectors and producing a scalar, linear in each vector separately. Matrix multiplication, viewed as a function of two matrices producing a third, is bilinear with respect to the appropriate vector space structures on the sets of matrices involved. The vector-covector pairing, taking a covector and a vector and returning a scalar by applying the covector to the vector, is likewise a bilinear map, and is in fact the most fundamental bilinear map associated with any vector space and its dual.

Bilinear Forms

A bilinear map whose output lies in the scalar field itself, rather than in some other vector space, is called a bilinear form. Bilinear forms may possess additional properties of interest, such as symmetry, in which swapping the two arguments leaves the value unchanged, or antisymmetry, in which swapping the two arguments negates the value. Symmetric bilinear forms underlie the definition of an inner product and, more generally, a metric, while antisymmetric bilinear forms are the rank-two case of alternating multilinear forms, which extend naturally into the exterior algebra.

From Bilinear Maps to the Tensor Product

The tensor product of two vector spaces is constructed precisely so that it satisfies a universal property connecting it to bilinear maps: for any bilinear map defined on the two original vector spaces, there is a unique linear map from their tensor product that reproduces the bilinear map's values when composed with the canonical bilinear map sending a pair of vectors to their elementary tensor. This universal property is what justifies treating the tensor product as the natural home for bilinear relationships, converting the two-argument, multiplicatively interacting behavior of a bilinear map into the single-argument, additive behavior of an ordinary linear map on a larger space.