pure tensors

In the tensor product V \otimes W, we defined a “pure tensor” to be an element of the form v \otimes w where v \in V, w \in W.

1. I warned you that not every element in V \otimes W is a pure tensor. Can you give an example of this? (Do keep in mind that something that does not immediately look like a pure tensor, such as (1 \otimes x) - (x \otimes 4x^2) + (2x \otimes x) - (1 \otimes 2x^2), might actually be a pure tensor. In this case it is (1+2x) \otimes (x-2x^2).)

2. Samantha asked a good question: How can we tell if two pure tensors v \otimes w and v' \otimes w' are equal to each other? Any thoughts? Is this easy? Is this hard?

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6 comments

  1. remifrazier

    For (2), it seems like we should be able to write any element of $V \times W$ as the sum (is it still a “sum” here, or something else?) of single-term tensors as in the first half of (1). Other than the distribution of coefficients, I think this would be a unique way to describe any element in the tensor product space. So we should be able to decompose any two pure tensors this way and compare the results fairly easily…although it’s not clear to me that the decomposition process would be easy, and I’m not sure that this wouldn’t get a lot harder depending on the space that $V$ and $W$ live in.

  2. remifrazier

    Then for (1), I think we can take a sum (“sum?”) of pure tensors which are elements of V \times W but which don’t “factor” well, like 1\otimes x + 1\otimes x^2 + 1\otimes x^3.

  3. remifrazier

    Ah! For (1) — consider v,v' \in V and w,w' in W, where v and v' are linearly independent, and w and w' are linearly independent. Then (v\otimes w + v' \otimes w')\in V \times W, but is not a pure tensor which can we written in the form a \otimes b.

  4. Remi, are you sure that your first example doesn’t factor as a pure tensor? (And can you – or anyone else – prove your second claim?)

  5. remifrazier

    I was sure, but now I’m not.

    For my first example: I think in principle, if I can take some v,v' \in V and w,w' \in W such that {v,v',w,w'} are all mutually linearly independent, then (v \otimes w + v' \otimes w') \in  V \times W cannot be written as a pure tensor. In effect, I’m claiming that the tensor products of the basis elements of V and W form the basis for V \times W, but I need to figure out how to prove that.

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