Editing Tensor (section)

=== Tensor products of vector spaces ===
The vector spaces of a [[tensor product]] need not be the same, and sometimes the elements of such a more general tensor product are called "tensors".  For example, an element of the tensor product space {{math|''V'' ⊗ ''W''}} is a second-order "tensor" in this more general sense,<ref name="Maia2011">{{cite book|first=M. D. |last=Maia|title=Geometry of the Fundamental Interactions: On Riemann's Legacy to High Energy Physics and Cosmology|url={{google books |plainurl=y |id=wEWw_vGBDW8C|page=48}}
|year=2011|publisher=Springer |isbn=978-1-4419-8273-5|page=48}}</ref> and an order-{{math|''d''}} tensor may likewise be defined as an element of a tensor product of {{math|''d''}} different vector spaces.<ref name="Hogben2013">{{cite book|url={{google books |plainurl=y |id=Er7MBQAAQBAJ|page=7}}|title=Handbook of Linear Algebra |publisher=CRC Press|year=2013|isbn=978-1-4665-0729-6|editor-last=Hogben|editor-first=Leslie|editor-link= Leslie Hogben |edition=2nd|pages=15–7}}</ref>  A type {{math|(''n'', ''m'')}} tensor, in the sense defined previously, is also a tensor of order {{math|''n'' + ''m''}} in this more general sense.  The concept of tensor product [[tensor product of modules|can be extended]] to arbitrary [[module over a ring|modules over a ring]].