Editing Vector space (section)

===Tensor product===
{{Main|Tensor product of vector spaces}}
The ''tensor product'' <math>V \otimes_F W,</math> or simply <math>V \otimes W,</math> of two vector spaces <math>V</math> and <math>W</math> is one of the central notions of [[multilinear algebra]] which deals with extending notions such as linear maps to several variables. A map <math>g : V \times W \to X</math> from the [[Cartesian product]] <math>V \times W</math> is called [[bilinear map|bilinear]] if <math>g</math> is linear in both variables <math>\mathbf{v}</math> and <math>\mathbf{w}.</math>  That is to say, for fixed <math>\mathbf{w}</math> the map <math>\mathbf{v} \mapsto g(\mathbf{v}, \mathbf{w})</math> is linear in the sense above and likewise for fixed <math>\mathbf{v}.</math>

[[Image:Universal tensor prod.svg|class=skin-invert-image|right|thumb|200px|[[Commutative diagram]] depicting the universal property of the tensor product]]
The tensor product is a particular vector space that is a ''universal'' recipient of bilinear maps <math>g,</math> as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called [[tensor]]s
<math display=block>\mathbf{v}_1 \otimes \mathbf{w}_1 + \mathbf{v}_2 \otimes \mathbf{w}_2 + \cdots + \mathbf{v}_n \otimes \mathbf{w}_n,</math>
subject to the rules{{sfn|Lang|2002|loc = ch. XVI.1}}
<math display=block>\begin{alignat}{6}
a \cdot (\mathbf{v} \otimes \mathbf{w}) ~&=~ (a \cdot \mathbf{v}) \otimes \mathbf{w} ~=~ \mathbf{v} \otimes (a \cdot \mathbf{w}), && ~~\text{ where } a \text{ is a scalar} \\
(\mathbf{v}_1 + \mathbf{v}_2) \otimes \mathbf{w} ~&=~ \mathbf{v}_1 \otimes \mathbf{w} + \mathbf{v}_2 \otimes \mathbf{w}  &&  \\
\mathbf{v} \otimes (\mathbf{w}_1 + \mathbf{w}_2) ~&=~ \mathbf{v} \otimes \mathbf{w}_1 + \mathbf{v} \otimes \mathbf{w}_2.  &&  \\
\end{alignat}</math>
These rules ensure that the map <math>f</math> from the <math>V \times W</math> to <math>V \otimes W</math> that maps a [[tuple]] <math>(\mathbf{v}, \mathbf{w})</math> to <math>\mathbf{v} \otimes \mathbf{w}</math> is bilinear. The universality states that given ''any'' vector space <math>X</math> and ''any'' bilinear map <math>g : V \times W \to X,</math> there exists a unique map <math>u,</math> shown in the diagram with a dotted arrow, whose [[function composition|composition]] with <math>f</math> equals <math>g:</math> <math>u(\mathbf{v} \otimes \mathbf{w}) = g(\mathbf{v}, \mathbf{w}).</math><ref>{{harvtxt|Roman|2005}}, Th. 14.3. See also [[Yoneda lemma]].</ref> This is called the [[universal property]] of the tensor product, an instance of the method—much used in advanced abstract algebra—to indirectly define objects by specifying maps from or to this object.