Timeline for What is the first interesting theorem in (insert subject here)?
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Jul 24, 2010 at 5:33 | comment | added | Victor Protsak | I think that various theorems about bases (every two bases have the same cardinality, every linearly independent set can be extended to a basis, and so on) are nontrivial and certainly come earlier. Why non-trivial? Because they may fail for modules over other rings, even if the module is free (but the ring may be noncommutative) or for f.g. modules over commutative rings (maximal linearly independent systems may have different cardinality, mathoverflow.net/questions/30066/…) | |
| May 23, 2010 at 20:52 | comment | added | Tom Smith | I've been teaching some linear algebra this term. It's really satisfying seeing pupils realise just what a powerful tool this theorem is. (Powerful, that is, in the context of first year lin alg...) | |
| Oct 25, 2009 at 2:08 | comment | added | GMRA | You could say that the rank nullity theorem is a direct consequence of the fact that all modules over fields are free (and hence projective). This may be a warped way of looking at things but I have to admit, thats how I remember it now :) | |
| Oct 24, 2009 at 23:04 | comment | added | S. Carnahan♦ | Do you mean the statement that a set is the union of the preimages of the image elements under a map? I think both statements are categorifications of the existence of addition, but linear categorifications are intrinsically more powerful. | |
| Oct 24, 2009 at 21:42 | comment | added | Kim Morrison | Although of course, this is just a categorification of the corresponding statement for finite sets... | |
| Oct 24, 2009 at 21:30 | history | answered | S. Carnahan♦ | CC BY-SA 2.5 |