Editing Product (mathematics) (section)

===Composition of linear mappings===
{{main|Function composition}}
A linear mapping can be defined as a function ''f'' between two vector spaces ''V'' and ''W'' with underlying field '''F''', satisfying<ref>{{cite book|last1=Clarke|first1=Francis|title=Functional analysis, calculus of variations and optimal control|date=2013|publisher=Springer|location=Dordrecht|isbn=978-1447148203|pages=9–10}}</ref>
:<math>f(t_1 x_1 + t_2 x_2) = t_1 f(x_1) + t_2 f(x_2), \forall x_1, x_2 \in V, \forall t_1, t_2 \in \mathbb{F}.</math>

If one only considers finite dimensional vector spaces, then
:<math>f(\mathbf{v}) = f\left(v_i \mathbf{b_V}^i\right) = v_i f\left(\mathbf{b_V}^i\right) = {f^i}_j v_i \mathbf{b_W}^j,</math>
in which '''b<sub>V</sub>''' and '''b<sub>W</sub>''' denote the [[Basis (linear algebra)|bases]] of ''V'' and ''W'', and ''v<sub>i</sub>'' denotes the [[Tensor#Definition|component]] of '''v''' on '''b<sub>V</sub>'''<sup>''i''</sup>, and [[Einstein notation|Einstein summation convention]] is applied.

Now we consider the composition of two linear mappings between finite dimensional vector spaces. Let the linear mapping ''f'' map ''V'' to ''W'', and let the linear mapping ''g'' map ''W'' to ''U''. Then one can get
:<math>g \circ f(\mathbf{v}) = g\left({f^i}_j v_i \mathbf{b_W}^j\right) = {g^j}_k {f^i}_j v_i \mathbf{b_U}^k.</math>

Or in matrix form:
:<math>g \circ f(\mathbf{v}) = \mathbf{G} \mathbf{F} \mathbf{v},</math>
in which the ''i''-row, ''j''-column element of '''F''', denoted by ''F<sub>ij</sub>'', is ''f<sup>j</sup><sub>i</sub>'', and ''G<sub>ij</sub>=g<sup>j</sup><sub>i</sub>''.

The composition of more than two linear mappings can be similarly represented by a chain of matrix multiplication.