Linear algebra: 013 Linear transformations I-Definition, Fundamental Theorem, Isomorphism

[wballardny]

Professor Roman, I have a general question, which relates to Theorem 2.14 introduced at 48:16. In particular, if \tau is a linear map from V to W, and B is a basis for V, should I think of $ \tau B $ (the image of B under \tau) as a set or a multiset? The statement of the theorem makes me think that it is a multiset, since for example there is a unique linear map from R^2 to R^1 such that (0, 1) goes to (1) and (1, 0) goes to (1). Letting B = {(1, 0), (0, 1)}, we see that if $ \tau B $ is a set, then $ \tau B = {(1)} $ is a basis for R^1. On the other hand, if $ \tau B $ is a multi set, then it seems that $ \tau B = {(1), (1)} $ is not a basis for R^1, since it is not linearly independent. I'm sorry if I missed an earlier clarification of this small detail, but thank you so much for your textbook and lectures. So far, they have been very insightful and fun to watch.

[kdr]

So standard lectures accompanied with your excellent books, it’s a pity that I have no chance to listen your lectures in class.