Solve Tensor Product: Expand Out Elements of Form (x_i \otimes 1)(1 \otimes x_j)

[Adorno]

Homework Statement

I don't need help with the main problem, just a calculation: I need to expand out elements of the form $(x_i \otimes 1)(1 \otimes x_j)$, etc.

Homework Equations

The Attempt at a Solution

Is there a property of the tensor product that I can use to expand out products like the ones above? I have a feeling that I can write $(x_i \otimes 1)(1 \otimes x_j) = ((x_i)(1) \otimes (1)(x_j)) = x_i \otimes x_j$, but I'm not 100% sure.\

 

[micromass]

Yes, that is correct. It follows because multiplication in the tensor product is defined as

$$(a\otimes b)(c\otimes d)=ac\otimes bd$$

 

[Arian.D]

Yes, that is correct. It follows because multiplication in the tensor product is defined as

$$(a\otimes b)(c\otimes d)=ac\otimes bd$$

I know this is an old thread, but why tensor product is defined like that? And what do you mean by tensor product in this case?