HW 2 #2

Can we extend linearly for the twist map T: C \otimes C \rightarrow C \otimes C ? Is this a linear map?

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About Crista

Like everyone in this class I love mathematics. Soy Colombiana and I know this to be true now more than ever. I look forward to the collaboration between SFSU and la Universidad de los Andes, so please message me :).

3 comments

  1. Yes, it’s supposed to be a linear map (otherwise its definition would be incomplete).

  2. Darij is right; I should have said that the twist map T: C \otimes C \rightarrow C \otimes C is meant to be linear.
    In fact (as is often done), I am being too brief in the definition of T. Remember that if I want to define a linear map f: C \otimes D \rightarrow L to a vector space L, I cannot define f(c \otimes d) arbitrarily on the pure tensors c \otimes d. What I really need to do is define a bilinear map f': C \times D \rightarrow L such that f' “descends” to f. I *can* define f': C \times D \rightarrow L by defining f'(c_i, d_j) arbitrarily for bases \{c_i\}_{i \in I} and \{d_j\}_{j \in J} of C and D respectively, and then extending bilinearly.
    I am not asking you to do this explicitly in this problem, but you should make sure that you know how to do it. (The first time you do this, it may seem a bit tricky. Once you do this kind of thing enough times, it achieves the status of “obvious”, and is therefore omitted.)

  3. Karen Walters

    Federico, is it sufficient to use commuting diagrams to prove 2bi, 2bii and 2biii as you did in your lecture?

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