Math 2.0

Linear and multilinear algebra; matrix theory Definition

Dual space

Builds on Linear formSpace of linear maps

Definition

Let FF be a field and let VV be a vector space over FF. The dual space of VV is the vector space V∗=Hom⁡F(V,F)V^* = \Hom_F(V, F) of all linear forms on VV. Its operations are pointwise: (f+g)(v)=f(v)+g(v)(f + g)(v) = f(v) + g(v) and (af)(v)=af(v)(af)(v) = a f(v).

Remarks

It is a special case of the space of linear maps. If dim⁡V=n\dim V = n is finite, then dim⁡V∗=n\dim V^* = n, see the dimension of the dual space. The elements of FnF^n act on column vectors by the dot product, and this identifies (Fn)∗(F^n)^* with FnF^n. For infinite-dimensional VV, the space V∗V^* is not isomorphic to VV.

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Vector space, definition, no evidence levelVector spacedefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear form, definition, no evidence levelLinear formdefinitionSpace of linear maps, definition, no evidence levelSpace of linear mapsdefinitionDual space, definition, no evidence levelDual spacedefinition
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  1. Vector space definition
  2. Linear map definition
  3. Linear form definition
  4. Space of linear maps definition
  5. Dual space definition

Where this leads

12 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Dual_space, Algebraic dual space
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DefinesDual space
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