# HW 2 #2

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

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 :).

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.