Editing Linear map (section)

==Vector space of linear maps==

The composition of linear maps is linear: if <math>f: V \to W</math> and <math display="inline">g: W \to Z</math> are linear, then so is their [[Relation composition|composition]] <math display="inline">g \circ f: V \to Z</math>. It follows from this that the [[class (set theory)|class]] of all vector spaces over a given field ''K'', together with ''K''-linear maps as [[morphism]]s, forms a [[category (mathematics)|category]].

The [[inverse function|inverse]] of a linear map, when defined, is again a linear map.

If <math display="inline">f_1: V \to W</math> and <math display="inline">f_2: V \to W</math> are linear, then so is their [[pointwise]] sum <math>f_1 + f_2</math>, which is defined by <math>(f_1 + f_2)(\mathbf x) = f_1(\mathbf x) + f_2(\mathbf x)</math>.

If <math display="inline">f: V \to W</math> is linear and <math display="inline">\alpha</math> is an element of the ground field <math display="inline">K</math>, then the map <math display="inline">\alpha f</math>, defined by <math display="inline">(\alpha f)(\mathbf x) = \alpha (f(\mathbf x))</math>, is also linear.

Thus the set <math display="inline">\mathcal{L}(V, W)</math> of linear maps from <math display="inline">V</math> to <math display="inline">W</math> itself forms a vector space over <math display="inline">K</math>,<ref>{{Harvard citation text |Axler|2015}} p. 52, § 3.3</ref> sometimes denoted <math display="inline">\operatorname{Hom}(V, W)</math>.<ref>{{Harvard citation text|Tu|2011}}, p. 19, § 3.1</ref> Furthermore, in the case that <math display="inline">V = W</math>, this vector space, denoted <math display="inline"> \operatorname{End}(V)</math>, is an [[associative algebra]] under [[composition of maps]], since the composition of two linear maps  is again a linear map, and the composition of maps is always associative. This case is discussed in more detail below.

Given again the finite-dimensional case, if bases have been chosen, then the composition of linear maps corresponds to the [[matrix multiplication]], the addition of linear maps corresponds to the [[matrix addition]], and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.

===Endomorphisms and automorphisms===
{{Main|Endomorphism|Automorphism}}
A linear transformation <math display="inline">f : V \to V</math> is an [[endomorphism]] of <math display="inline">V</math>; the set of all such endomorphisms <math display="inline">\operatorname{End}(V)</math>  together with addition, composition and scalar multiplication as defined above forms an [[associative algebra]] with identity element over the field <math display="inline">K</math> (and in particular a [[ring (algebra)|ring]]). The multiplicative identity element of this algebra is the [[identity function|identity map]] <math display="inline">\operatorname{id}: V \to V</math>.

An endomorphism of <math display="inline">V</math> that is also an [[isomorphism]] is called an [[automorphism]] of <math display="inline">V</math>. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of <math display="inline">V</math> forms a [[group (math)|group]], the [[automorphism group]] of <math display="inline">V</math> which is denoted by <math display="inline">\operatorname{Aut}(V)</math> or <math display="inline">\operatorname{GL}(V)</math>. Since the automorphisms are precisely those [[endomorphisms]] which possess inverses under composition, <math display="inline">\operatorname{Aut}(V)</math> is the group of [[Unit (ring theory)|units]] in the ring <math display="inline">\operatorname{End}(V)</math>.

If <math display="inline">V</math> has finite dimension <math display="inline">n</math>, then <math display="inline"> \operatorname{End}(V)</math> is [[isomorphism|isomorphic]] to the [[associative algebra]] of all <math display="inline">n \times n</math> matrices with entries in <math display="inline">K</math>. The automorphism group of <math display="inline">V</math> is [[group isomorphism|isomorphic]] to the  [[general linear group]] <math display="inline">\operatorname{GL}(n, K)</math> of all <math display="inline">n \times n</math> invertible matrices with entries in <math display="inline">K</math>.