Timeline for Surjectivity of bilinear forms.
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| May 19, 2010 at 17:53 | history | edited | Xandi Tuni | CC BY-SA 2.5 |
added 5 characters in body
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| May 19, 2010 at 17:49 | vote | accept | Xandi Tuni | ||
| May 19, 2010 at 2:10 | comment | added | Victor Protsak | @Theo: Not every element of the tensor product space is decomposable, mathoverflow.net/questions/23478/… | |
| May 19, 2010 at 1:54 | answer | added | Bjorn Poonen | timeline score: 11 | |
| May 19, 2010 at 1:50 | comment | added | Bjorn Poonen | @Theo: A bilinear map induces a map on the tensor product, but they are not the same. In particular, the latter can be surjective even if the former is not. | |
| May 18, 2010 at 23:17 | comment | added | Theo Johnson-Freyd | So, since I think the question is trivial, I probably misinterpreted it. Or maybe you mean to be asking about extensions of rings? Then it is certainly nontrivial. | |
| May 18, 2010 at 23:16 | comment | added | Theo Johnson-Freyd | (con't) Indeed: for finite-dimensional spaces we can simply pick bases and then test whether $\beta$ is surjective by some elementary row operations, and those operations are defined over the minimum field that contains the matrix coefficients for $\beta$. In particular, the whole point of extending by scalars is that you don't change the matrix coefficients. So I think the answer to Question A is "all field extensions" and to Question B is "yes", but not for deep reasons. | |
| May 18, 2010 at 23:14 | comment | added | Theo Johnson-Freyd | Maybe I have misunderstood your question, so I will leave this as a comment. A bilinear map $\beta_K : U\times V\to W$ is (by universal property) the same as a linear map $\beta_K: U\otimes_K V \to W$. Also note that the extension-by-scalars of a tensor product is the tensor product of extensions-by-scalars. Anyway, then I think that if $\beta: X \to W$ is surjective, then any field extension is surjective, and conversely. (con't) | |
| May 18, 2010 at 21:00 | history | edited | Xandi Tuni | CC BY-SA 2.5 |
added first paragraph
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| May 18, 2010 at 19:53 | history | asked | Xandi Tuni | CC BY-SA 2.5 |