Math 2.0

Linear and multilinear algebra; matrix theory Definition

Space of linear maps

Builds on Linear map

Definition

Let FF be a field and let VV and WW be vector spaces over FF. The space of linear maps Hom⁡F(V,W)\Hom_F(V, W), also written L(V,W)L(V, W), is the set of all linear maps V→WV \to W. It has the operations (S+T)(v)=S(v)+T(v),(aT)(v)=a T(v)(S + T)(v) = S(v) + T(v), \qquad (aT)(v) = a\, T(v) for S,T∈Hom⁡F(V,W)S, T \in \Hom_F(V, W), a∈Fa \in F and v∈Vv \in V. It is a vector space over FF, with the zero map as zero vector.

Remarks

For V=FnV = F^n and W=FmW = F^m, the matrix map is an isomorphism Hom⁡F(Fn,Fm)≅Mm×n(F)\Hom_F(F^n, F^m) \cong M_{m \times n}(F). The dimension of the space of linear maps is dim⁡V⋅dim⁡W\dim V \cdot \dim W in finite dimensions. For W=FW = F, it is the dual space of VV, and for W=VW = V, it is the space of endomorphisms of VV.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Linear_map, Vector space of linear maps
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