Given nontrivial one-dimensional vector bundle $E$ on circle (which is of course Möbius strip). I need to find out if $E \oplus E \oplus ... \oplus E$ (n times) would be trivial or not (my guess is that it never happens) and the same question about $E \otimes E \otimes ... \otimes E$.
In the second case I can first of all notice that $E \otimes E$ is also one-dimensional and then we can look at transition maps. Let it be just multiplying by $-1$ (trivialization is considered in stereographic projections charts on circle). Well, then in $E \otimes E$ it will be $(-1) \otimes (-1) = 1 \otimes 1$ which makes this bundle into trivial one. For $E \otimes E \otimes E$ we obtain $-1 \otimes 1 \otimes 1= - id$ and so this bundle is not trivial and so on. We will obtain cylinder for even $n$ and Möbius strip for odd. Am I right?
I would appreciate some explanation regarding to first question and indication of mistakes (if there are) in my solution.