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 V to zero (the “annihilator” of V 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*)*=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.