Math 55a: Duality basics

The dual of an F-vector space V is the F-vector space Hom(V,F) of linear maps from V to F, often called functionals on V. We'll usually denote the dual of V by V^*. If W is the direct sum of U and V Then W^* is the direct sum of U^* and V^*; for instance, the copy of U^* in W^* consists of the functionals mapping all vectors in U to zero (the ``annihilator'' of U in W^*). The dual of a quotient space V/U is naturally a subspace of V^*, namely the annihilator of U in V^*.

If V has finite dimension n then so does V^*. The dimension of the annihilator of U in V^* then equals the codimension of U in V. If we choose a basis of V, and use it to identify elements of V with ``column vectors'' of length n, then elements of V^* correspond to ``row vectors'' of the same length. There is in general no canonical isomorphism between V and V^*, even if V is finite dimensional. We do, however, have a canonical map from V to its second dual V^**, which is an isomorphism when V is finite dimensional. The key idea is that v^*(v) is ``bilinear'': not only is it a linear function of v for each choice of v^*, but it's also a linear function of v^* for each choice of v -- so each v may be considered as a functional on V^*, that is, as an element of V^**!

If T is a linear transformation from V to W then we get a linear transformation T^* from W^* to V^*, called the adjoint of T and denoted by T^*, defined as follows: T^*(w^*) is the element of V^* taking any v to w^*(T(v)). The adjoint is itself a linear map from Hom(V,W) to Hom(W^*,V^*). This map is an isomorphism when V and W are finite dimensional. In that case, if we choose bases for V and W then the matrix of T^* with respect to the dual bases is the transpose of the matrix of T with respect to our bases for V,W. Naturally T^** is identified with T under the canonical maps that identify V^** and W^** with V and W.