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I'm just learning about tensors and the tensor product, and have been given examples for tensors such as $a \equiv x_1^2+x_2^2$ being one because it is the contraction of the tensorproduct of $\{x_i\}$ with itself, $\{x_i\}$ being a rank-1 tensor in 2 dimensions. I have only encountered the contraction as the trace of matrix, so how does it apply to this case? Also, can I understand $\{x_i\}$ as a $2 \times 2$ matrix and for example show that a new variable $b \equiv x_1+x_2$ is a tensor because it is the trace of the tensor $\{x_i\}$?

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  • $\begingroup$ A rank one tensor is of type $(1,0)$ or of type $(0,1)$. As the contraction of a $(p,q)$ tensor gives birth to a $(p-1,q-1)$ tensor, here it would have no sense because it would have a covariance or contravariance $-1$ $\endgroup$ – DIdier_ May 27 at 13:26

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