Linear Algebra: 011 Vector Spaces VI Bases and Dimension

[unixguru]

Nice lecture! Regarding your point about uncountable dimension vector spaces being difficult to work with as finding an explicit basis is hard. What useful conclusions could we make using the mere existence of an uncountable basis that we would not been able to make otherwise? You have shown one useful fact that complements always exist. Are there more?

[wballardny]

Professor Roman, I have a question about the equality at 55:50. In particular, the text appears to claim that $ S \cap T = \{0\}$, which seems at odds with our wish for the theorem to apply to sums of subspaces which are not necessarily essentially disjoint. Would it work instead to claim that $a=-(b+c) \in S \cap T$ implies that $a$ belongs to both the space generated by the a-vectors and the space generated by the b-vectors? Thank you for your tremendous textbook and very insightful lectures.