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.
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 $i=1,2,\ldots,m$.
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.
Best Answer
Vandermonde determinants + Cramer's rule = Lagrange interpolation.
(EDIT: Also, there is a qualitative version of the above identity: just from knowing that the Vandermonde determinant is non-vanishing when the $x_i$ are distinct, one can already deduce that polynomial interpolation is theoretically possible, though to get the precise formula one still needs to go through the above identity.)