Editing Linear algebra (section)

===Multilinear algebra and tensors===
{{cleanup|section|reason=The dual space is considered above, and the section must be rewritten to give an understandable summary of this subject|date=September 2018}}
In [[multilinear algebra]], one considers multivariable linear transformations, that is, mappings that are linear in each of several different variables. This line of inquiry naturally leads to the idea of the [[dual space]], the vector space {{math|''V*''}} consisting of linear maps {{math|''f'' : ''V'' → ''F''}} where ''F'' is the field of scalars. Multilinear maps {{math|''T'' : ''V<sup>n</sup>'' → ''F''}} can be described via [[tensor product]]s of elements of {{math|''V*''}}.

If, in addition to vector addition and scalar multiplication, there is a bilinear vector product {{math|''V'' × ''V'' → ''V''}}, the vector space is called an [[Algebra over a field|algebra]]; for instance, associative algebras are algebras with an associate vector product (like the algebra of square matrices, or the algebra of polynomials).