Editing Multilinear form (section)

== Tensor product ==
Given a <math>k</math>-tensor <math>f\in\mathcal{T}^k(V)</math> and an <math>\ell</math>-tensor <math>g\in\mathcal{T}^\ell(V)</math>, a product <math>f\otimes g\in\mathcal{T}^{k+\ell}(V)</math>, known as the '''tensor product''', can be defined by the property

: <math>(f\otimes g)(v_1,\ldots,v_k,v_{k+1},\ldots, v_{k+\ell})=f(v_1,\ldots,v_k)g(v_{k+1},\ldots, v_{k+\ell}),</math>

for all <math>v_1,\ldots,v_{k+\ell}\in V</math>.  The [[tensor product]] of multilinear forms is not commutative; however it is bilinear and associative:

: <math>f\otimes(ag_1+bg_2)=a(f\otimes g_1)+b(f\otimes g_2)</math>, <math>(af_1+bf_2)\otimes g=a(f_1\otimes g)+b(f_2\otimes g),</math>

and

: <math>(f\otimes g)\otimes h=f\otimes (g\otimes h).</math>

If <math>(v_1,\ldots, v_n)</math> forms a basis for an <math>n</math>-dimensional vector space <math>V</math> and <math>(\phi^1,\ldots,\phi^n)</math> is the corresponding [[dual basis]] for the [[dual space]] <math>V^*=\mathcal{T}^1(V)</math>, then the products <math>\phi^{i_1}\otimes\cdots\otimes\phi^{i_k}</math>, with <math>1\le i_1,\ldots,i_k\le n</math> form a basis for <math>\mathcal{T}^k(V)</math>.  Consequently, <math>\mathcal{T}^k(V)</math> has dimension <math>n^k</math>.