# Tensor

• December 6th 2009, 03:32 PM
smith09
Tensor
Prove that every element $v\in (\mathbb{C}^{2})^{\otimes 3}$ is the sum of two pure tensors $u_{1}\otimes u_{2}\otimes u_{3}$., where $u_{1},u_{2},u_{3}\in\mathbb{C}^{2}$ .

• December 7th 2009, 12:26 AM
NonCommAlg
Quote:

Originally Posted by smith09
Prove that every element $v\in (\mathbb{C}^{2})^{\otimes 3}$ is the sum of two pure tensors $u_{1}\otimes u_{2}\otimes u_{3}$., where $u_{1},u_{2},u_{3}\in\mathbb{C}^{2}$ .

you're sure that two is not three then? the reason for asking this is that, in general, for any 2-dimensional vector space $V$ over a field $F,$ the "maximum rank" of $V^{\otimes 3}$ is 3 and not 2, i.e. every

element of $V^{\otimes 3}$ is a sum of at most 3 simple tensors.
• December 7th 2009, 02:26 AM
smith09
Thks, but the question requires 2, not 3. They give some hints that consider $v$ as a linear map from $\mathbb{C}^2$ to $M_2$ (group of 2x2 matrices) (WHY? and HOW?P) and consider two possibilities of the dimension of the image (1 or 2). But I really dont get it.
• December 7th 2009, 12:42 PM
smith09
Oh, sorry, $v$ here must be in a dense open subset.
• December 15th 2009, 11:24 AM
smith09
Is there any ideal then? I think the first case (1 dim) is not so hard. But the second case is ...not easy.