Timeline for Proof that bases etc. exist in early linear algebra course?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Oct 7, 2010 at 11:01 | comment | added | Andrew Stacey | Drvitek: Did no-one provide chairs for the polar bears? (Seriously, I wasn't happy with the picture of the great ship Linear Algebra heading for the depths, and preferred the picture of a body of mathematics which is mostly (for the students) under water.) | |
| Oct 7, 2010 at 10:36 | comment | added | dvitek | I was under the impression that the deck chairs were on the Titanic, not the iceberg. | |
| Oct 6, 2010 at 16:09 | comment | added | Andrew Stacey | Mariano: apparently, I encouraged Bad Teaching here: mathoverflow.net/questions/40082/… | |
| Oct 6, 2010 at 16:07 | comment | added | Andrew Stacey | (ctd) The key step is the fact that Gaussian Elimination is actually a theorem about dimension. It says that if $T \colon \mathbb{R}^n \to \mathbb{R}^m$ is an injection then $n \le m$, and so on. Of course, to a certain extent, it's rearranging deck chairs on the iceberg of Linear Algebra, but the trick is to make it so that each step along the way is notable for its own sake and not just as a step towards some Big Result (which is never quite as Big or Resulty as the build-up promises). | |
| Oct 6, 2010 at 16:05 | comment | added | Andrew Stacey | David, actually they are. There's various ways to do them, but one which shows the power of the technique is to observe that once these facts are established for $\mathbb{R}^n$ then they automatically hold for all finite dimensional vector spaces. And in $\mathbb{R}^n$ I can use whatever extra structure I like! For example, I can use the standard inner product to split quotient mappings. Thus both become questions about subspaces. And that's easy once I prove that if a space is not finite dimensional then it is infinite dimensional (ie contains a copy of each $\mathbb{R}^n$). (ctd) | |
| Oct 6, 2010 at 13:45 | comment | added | Mariano Suárez-Álvarez | When did you encourage Bad Teaching Last Time? :) | |
| Oct 6, 2010 at 13:43 | comment | added | David E Speyer | It seems to me that you should also prove at some point that a subspace of a finite dimensional vector space is finite dimensional, and a quotient of a finite dimensional vector space is finite dimensional. These facts are not obvious given the way you have set up the definitions. | |
| Oct 6, 2010 at 13:27 | history | answered | Andrew Stacey | CC BY-SA 2.5 |