Dual Vector Space

The dual vector space to a real vector space V is the vector space of linear functions f:V->R, denoted V^*. In the dual of a complex vector space, the linear functions take complex values.

In either case, the dual vector space has the same dimension as V. Given a vector basis v_1, ..., v_n for V there exists a dual basis for V^*, written v_1^*, ..., v_n^*, where v_i^*(v_j)=delta_(ij) and delta_(ij) is the Kronecker delta.

Another way to realize an isomorphism with V is through an inner product. A real vector space can have a symmetric inner product <,> in which case a vector v corresponds to a dual element by f_v(w)=<w,v>. Then a basis corresponds to its dual basis only if it is an orthonormal basis, in which case v_i^*=f_(v_i). A complex vector space can have a Hermitian inner product, in which case f_v(w)=<w,v> is a conjugate-linear isomorphism of V with V^*, i.e., f_(alphav)=alpha^_f_v.

Dual vector spaces can describe many objects in linear algebra. When V and W are finite dimensional vector spaces, an element of the tensor product V^* tensor W, say suma_(ij)v_j^* tensor w_i, corresponds to the linear transformation T(v)=suma_(ij)v_j^*(v)w_i. That is, V^* tensor W=Hom(V,W). For example, the identity transformation is v_1 tensor v_1^*+...+v_n tensor v_n^*. A bilinear form on V, such as an inner product, is an element of V^* tensor V^*.

See also

Bilinear Form, Dual Normed Space, Generalized Function, Linear Functional, Matrix, Self-Dual, Vector Basis, Vector Space

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Dual Vector Space." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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