Editing Linear map (section)

==Definition and first consequences==

Let <math>V</math> and <math>W</math> be vector spaces over the same [[Field (mathematics)|field]] <math>K</math>. 
A [[function (mathematics)|function]] <math>f: V \to W</math> is said to be a ''linear map'' if for any two vectors <math display="inline">\mathbf{u}, \mathbf{v} \in V</math> and any scalar <math>c \in K</math> the following two conditions are satisfied:

* [[Additive map|Additivity]] / operation of addition <math display=block>f(\mathbf{u} + \mathbf{v}) = f(\mathbf{u}) + f(\mathbf{v})</math>
* [[Homogeneous function|Homogeneity]] of degree 1 / operation of scalar multiplication <math display=block>f(c \mathbf{u}) = c f(\mathbf{u})</math>

Thus, a linear map is said to be ''operation preserving''. In other words, it does not matter whether the linear map is applied before (the right hand sides of the above examples) or after (the left hand sides of the examples) the operations of addition and scalar multiplication.

By [[Addition#Associativity|the associativity of the addition operation]] denoted as +, for any vectors <math display="inline"> \mathbf{u}_1, \ldots, \mathbf{u}_n \in V</math> and scalars <math display="inline">c_1, \ldots, c_n \in K,</math> the following equality holds:<ref>{{harvnb|Rudin|1991|page=14}}. Suppose now that {{mvar|X}} and {{mvar|Y}} are vector spaces ''over the same scalar field''. A mapping <math display="inline">\Lambda: X \to Y</math> is said to be ''linear'' if <math display="inline"> \Lambda(\alpha \mathbf x + \beta \mathbf y) = \alpha \Lambda \mathbf x + \beta \Lambda \mathbf y</math> for all <math display="inline">\mathbf x, \mathbf y \in X</math> and all scalars <math display="inline">\alpha</math> and <math display="inline">\beta</math>. Note that one often writes <math display="inline">\Lambda \mathbf x</math>, rather than <math display="inline">\Lambda(\mathbf x)</math>, when <math display="inline"> \Lambda</math> is linear.</ref><ref>{{harvnb|Rudin|1976|page=206}}. A mapping {{mvar|A}} of a vector space {{mvar|X}} into a vector space {{mvar|Y}} is said to be a ''linear transformation'' if: <math display="inline">A\left(\mathbf{x}_1 + \mathbf{x}_2\right) = A\mathbf{x}_1 + A\mathbf{x}_2,\  A(c\mathbf{x}) = c A\mathbf{x}</math> for all <math display="inline">\mathbf{x}, \mathbf{x}_1, \mathbf{x}_2 \in X</math> and all scalars {{mvar|c}}. Note that one often writes <math display="inline">A\mathbf{x}</math> instead of <math display="inline">A(\mathbf {x})</math> if {{mvar|A}} is linear.</ref>
<math display="block">f(c_1 \mathbf{u}_1 + \cdots + c_n \mathbf{u}_n) = c_1 f(\mathbf{u}_1) + \cdots + c_n f(\mathbf{u}_n).</math>  Thus a linear map is one which preserves [[linear combination]]s.

Denoting the zero elements of the vector spaces <math>V</math> and <math>W</math> by <math display="inline">\mathbf{0}_V</math> and <math display="inline">\mathbf{0}_W</math> respectively, it follows that <math display="inline">f(\mathbf{0}_V) = \mathbf{0}_W.</math> Let <math>c = 0</math> and <math display="inline">\mathbf{v} \in V</math> in the equation for homogeneity of degree 1:
<math display="block">f(\mathbf{0}_V) = f(0\mathbf{v}) = 0f(\mathbf{v}) = \mathbf{0}_W.</math>

A linear map <math>V \to K</math> with <math>K</math> viewed as a one-dimensional vector space over itself is called a [[linear functional]].<ref>{{harvnb|Rudin|1991|page=14}}. 
Linear mappings of {{mvar|X}} onto its scalar field are called ''linear functionals''.</ref>

These statements generalize to any left-module <math display="inline">{}_R M</math> over a ring <math>R</math> without modification, and to any right-module upon reversing of the scalar multiplication.