User zhaoliang - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:59:36Z http://mathoverflow.net/feeds/user/10322 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43538/wonderful-applications-of-the-vandermonde-determinant Wonderful applications of the Vandermonde determinant zhaoliang 2010-10-25T16:17:58Z 2012-09-21T06:53:21Z <p>This semester I am assisting my mentor teaching a first-year undergraduate course on linear algebra in Peking University, China. And now we have come to the famous Vandermonde determinant, which has many useful applications. I wonder if there are some applications of the Vandermonde determinant that are suitable for students without much math background. </p> <p>For example, using the Vandermonde determinant, we can prove that a vector space $V$ over a field $F$ of characteristic 0 cannot be expressed as a finite union of its nontrivial subspaces, i.e., there do not exist subspaces $V_1,\ldots,V_m$ that satisfy $$V_1\cup \cdots\cup V_m=V,$$ where $V_i\ne {0}$ and $V_i\ne V$ for all $i=1,2,\ldots,m$.</p> <p>This can be proved as follows: choose $v_1,\ldots,v_n$ as a basis of $V$, and consider the infinite series $$\alpha_i = v_1 + iv_2+\cdots+i^{n-1}v_n.$$ Using our knowledge of the Vandermonde determinant, one can show that every subset of the $\alpha$'s having $n$ vectors in it consists of a basis of $V$, hence each of the $V_i$'s can contain at most $n-1$ of the $\alpha$'s in it, so there must be infinitely many $\alpha$'s not contained in any of the $V_i$'s.</p> http://mathoverflow.net/questions/43538/wonderful-applications-of-the-vandermonde-determinant/43553#43553 Comment by zhaoliang zhaoliang 2010-10-25T17:54:58Z 2010-10-25T17:54:58Z When I was typing this item, I had Lagrange interpolation in my mind :P I think there should be some applications in combinatorics or symmetric function theory http://mathoverflow.net/questions/43538/wonderful-applications-of-the-vandermonde-determinant/43548#43548 Comment by zhaoliang zhaoliang 2010-10-25T17:50:16Z 2010-10-25T17:50:16Z this reminds me the Linderman-Weierstrass theorem on the transcendence of e. http://mathoverflow.net/questions/43538/wonderful-applications-of-the-vandermonde-determinant/43543#43543 Comment by zhaoliang zhaoliang 2010-10-25T17:18:23Z 2010-10-25T17:18:23Z <a href="http://gigapedia.com/items:links?eid=EgPsEJFBgjqQ3mBP5hh7EwHptTI%2FHyCKvE1rFBP%2FhDo%3D" rel="nofollow">gigapedia.com/&hellip;</a> http://mathoverflow.net/questions/43538/wonderful-applications-of-the-vandermonde-determinant Comment by zhaoliang zhaoliang 2010-10-25T16:35:00Z 2010-10-25T16:35:00Z yes, but here the set has an extra property: each subset with n elements is a base of $V$, which is a bit more interesting.
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