Hello! Im currently reading the analysis of tensors and have now encountered the tensorproduct, [tex] \otimes [/tex].(adsbygoogle = window.adsbygoogle || []).push({});

I am wondering about the statement that every vector in: [tex]V \otimes W [/tex] (with the basis (v_i) and (w_i)) can be written as a linear combination of the basis: [tex]v_i \otimes w_i ,[/tex] but not in general as an element of the form:[tex] v \otimes w,[/tex] where w and v are elements i W and V.

Which elements can I not reach by the second way? If we set V and W to R^3 it looks like we are comparing a 6-dimensional space to a 9-dimensional space (true?), in that case does it have something to do with the symmetric or antisymmetric components of [tex]V \otimes W [/tex] that can not be reached by [tex] v \otimes w[/tex]?

I am thankful for all help possible. :)

Best Regards

Kontilera

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# Direct product between vectorspaces

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