For any $n$ distinct points $x_1,x_2 , \ldots , x_n$ on the real line show that the matrix $M$ where $M(i,j) = e^{\lambda_j x_i} $ has non-zero determinant where $\lambda_1 \lt \lambda_2 \lt \ldots \lt \lambda_n \in \mathbb{R}$ are fixed constants.
I am able to show this for $n=1$(duh...) and $n=2$. Is this an inductive proof?
Just in case it helps: to see that the OP's matrix $M$ has nonzero determinant using the argument in Noam Elkies' argument above, consider
$$\begin{equation*} \mathrm{det}\; \begin{pmatrix} e^{\lambda_1 x_1} & e^{\lambda_2 x_1} & \cdots & e^{\lambda_n x_1}\\\\ e^{\lambda_1 x_2} & e^{\lambda_2 x_2} & \cdots & e^{\lambda_n x_2}\\\\ \vdots & \vdots & & \vdots\\\\ e^{\lambda_1 x} & e^{\lambda_2 x} & \cdots & e^{\lambda_n x} \end{pmatrix} \end{equation*} $$
as an exponential polynomial $f(x) = \sum_{k=1}^n a_k e^{\lambda_k x}$. This has roots at $x = x_1, x_2, \ldots, x_{n-1}$. On the other hand, Elkies argued (see mathoverflow.net/questions/83999) that an exponential polynomial with $n$ terms has at most $n-1$ real roots. (We actually need only the weaker claim that $f(x)$ has at most $n-1$ distinct real roots. This follows by induction, where the inductive step involves Rolle's theorem applied to the derivative of the exponential polynomial $e^{-\lambda_n x}f(x)$, which has $n-1$ terms.) Thus $f(x)$ must be nonzero at $x = x_n$, as desired.
EDIT: OOPS! The solution below is FALSE. My apologies for wasting everyone's time (particularly those among you who already were aware of Schur functions) with this scrible. I'll let it stay around because maybe this thread will one day be renamed "how not to show a certain determinant..." and maybe there is something to be learned from it.
For the record, the mistake is to claim that "multiplying $\lambda_1$, $\lambda_2$, ..., $\lambda_n$ by their common denominator $p$ is easily compensated by replacing the positive reals $y_1$, $y_2$, ..., $y_n$ by the (equally positive) reals $y_1^{1/p}$, $y_2^{1/p}$, ..., $y_n^{1/p}$". This is true for the original question, but not for the more subtle (1). Sorry again!
For the sake of pushing algebraic combinatorics (specifically Schur polynomials), let me present a different proof of the nonvanishing of the determinant in question. The idea of this proof is taken from my Jan 7, 2013 comment on the original question, but it has been tweaked to work in the general case.
We want to prove that $\det\left(\left(e^{\lambda_j x_i}\right)_{1\leq i,j\leq n}\right)\neq 0$ for any $n$ distinct reals $x_1$, $x_2$, ..., $x_n$ and any $n$ distinct reals $\lambda_1$, $\lambda_2$, ..., $\lambda_n$.
Let us WLOG assume that $x_1 > x_2 > ... > x_n$ and $\lambda_1 > \lambda_2 > ... > \lambda_n$ (because interchanging the $x_i$ or the $\lambda_i$ boils down to row resp. column swaps on the matrix whose determinant we are concerned with).
Let us denote $y_i = e^{x_i}$ for every $i\in\left\lbrace 1,2,...,n\right\rbrace$. Then, $y_1$, $y_2$, ..., $y_n$ are $n$ positive reals satisfying $y_1 > y_2 > ... > y_n$, and we must prove that $\det\left(\left(y_i^{\lambda_j}\right)_{1\leq i,j\leq n}\right)\neq 0$ (since $e^{\lambda_j x_i} = \left(e^{x_i}\right)^{\lambda_j} = y_i^{\lambda_j}$ for any $i$ and $j$).
We can WLOG assume that $\lambda_1$, $\lambda_2$, ..., $\lambda_n$ are all nonnegative, since we could replace the whole $n$-tuple $\left(\lambda_1,\lambda_2,...,\lambda_n\right)$ by $\left(\lambda_1+m,\lambda_2+m,...,\lambda_n+m\right)$ for a sufficiently large real $m$ without changing much about our determinant (namely, the determinant would merely gain a multiplicative factor of $y_1^m y_2^m ... y_n^m$). So assume this.
We will actually prove that
(1) $\det\left(\left(y_i^{\lambda_j}\right)_{1\leq i,j\leq n}\right) \geq \prod\limits_{1\leq i < j \leq n} \left(y_i-y_j\right) \cdot \prod\limits_{k=1}^n y_k^{\lambda_k - n + k}$.
Once this is proven, it will follow that $\det\left(\left(y_i^{\lambda_j}\right)_{1\leq i,j\leq n}\right) > 0$ (because the right hand side of (1) is positive) and thus $\det\left(\left(y_i^{\lambda_j}\right)_{1\leq i,j\leq n}\right) \neq 0$, which is exactly what we need to prove.
So it remains to prove (1) for any $n$ positive reals $y_1$, $y_2$, ..., $y_n$ satisfying $y_1 > y_2 > ... > y_n$, and any $n$ nonnegative reals $\lambda_1$, $\lambda_2$, ..., $\lambda_n$ satisfying $\lambda_1 > \lambda_2 > ... > \lambda_n$.
Note that both sides of the inequality (1) are continuous as functions in $\lambda_1$, $\lambda_2$, ..., $\lambda_n$. Hence, we can WLOG assume that $\lambda_1$, $\lambda_2$, ..., $\lambda_n$ are nonnegative rationals (because the inequality is non-strict, and the set of strictly increasing $n$-tuples of nonnegative rationals is dense in the set of strictly increasing $n$-tuples of nonnegative reals). Assuming this, we can go further and assume WLOG that $\lambda_1$, $\lambda_2$, ..., $\lambda_n$ are nonnegative integers, because multiplying $\lambda_1$, $\lambda_2$, ..., $\lambda_n$ by their common denominator $p$ is easily compensated by replacing the positive reals $y_1$, $y_2$, ..., $y_n$ by the (equally positive) reals $y_1^{1/p}$, $y_2^{1/p}$, ..., $y_n^{1/p}$. So assume this, and let $\mu$ denote the sequence $\left(\lambda_1+n-1, \lambda_2+n-2, ..., \lambda_n+n-n\right)$. This sequence $\mu$ is a partition (in the sense of algebraic combinatorics), i. e., a finite weakly decreasing sequence of nonnegative integers.
Now, for any sequence $\kappa = \left(\kappa_1,\kappa_2,...,\kappa_n\right)$ of nonnegative integers, let $a_{\kappa}$ denote the determinant $\det\left(\left(y_i^{\kappa_j}\right)_{1\leq i,j\leq n}\right)$. Let $\rho$ be the sequence $\left(n-1,n-2,...,0\right)$. Then, $a_{\rho} = \prod\limits_{1\leq i < j \leq n} \left(y_i-y_j\right)$ (by Vandermonde's determinant) while (using the notation $\mu+\rho$ for the termwise sum of the sequences $\mu$ and $\rho$) we have $a_{\mu+\rho} = \det\left(\left(y_i^{\lambda_j}\right)_{1\leq i,j\leq n}\right)$ (since $\mu+\rho$ is the sequence $\left(\lambda_1,\lambda_2,...,\lambda_n\right)$). Hence, (1) rewrites as
$a_{\mu+\rho} \geq a_{\rho} \cdot \prod\limits_{k=1}^n y_k^{\lambda_k - n + k}$.
Since $a_{\rho} = \prod\limits_{1\leq i < j \leq n} \left(y_i-y_j\right) > 0$, this is equivalent to
(2) $\dfrac{a_{\mu+\rho}}{a_{\rho}} \geq \prod\limits_{k=1}^n y_k^{\lambda_k - n + k}$.
But it is known (e. g., Corollary 2.37 in Victor Reiner, Hopf algebras in combinatorics) that $\dfrac{a_{\mu+\rho}}{a_{\rho}}$ equals the Schur polynomial $s_{\mu}$ evaluated at $\left(y_1,y_2,...,y_n\right)$. Thus,
(3) $\dfrac{a_{\mu+\rho}}{a_{\rho}} = s_{\mu}\left(y_1,y_2,...,y_n\right) = \sum\limits_{T} \prod\limits_{k=1}^n y_k^{\text{number of }k\text{'s in }T}$,
where the $T$ on the right hand side runs over all semistandard (i. e., column-strict) Young tableaux of shape $\mu$ with entries in $\left\lbrace 1,2,...,n\right\rbrace$. One such tableau is obtained by filling each cell in row $k$ with the number $k$, for every $k \in \left\lbrace 1,2,...,n\right\rbrace$ (where the numbering of rows starts with $1$). This tableau contributes one addend to the sum on the right hand side of (3), and this addend is $\prod\limits_{k=1}^n y_k^{\lambda_k - n + k}$ (because the length of the $k$-th row is $\lambda_k - n + k$, and the $k$'s in the tableau are exactly the entries of the $k$-th row). Since all the other addends on the right hand side of (3) are nonnegative (being monomials in $y_1$, $y_2$, ..., $y_n$ with coefficient $1$), this yields
$\dfrac{a_{\mu+\rho}}{a_{\rho}} \geq \prod\limits_{k=1}^n y_k^{\lambda_k - n + k}$.
But this is exactly (2). Since we know that (2) is equivalent to (1), this completes the proof of (1), and thus solves the problem.
Remark: Why did we take the detour through (1) rather than confine ourselves to proving the (weaker but sufficient) inequality
(4) $\det\left(\left(y_i^{\lambda_j}\right)_{1\leq i,j\leq n}\right) > 0$ ?
Because (1) is a non-strict inequality, whereas (4) is strict. When proving a strict inequality, it is not enough to prove it on a dense subset of its domain, even if it is continuous; for example, Vasile Cîrtoaje's brainteaser $\left(x^2+y^2+z^2\right)^2\geq 3\left(x^3y+y^3z+z^3x\right)$ (an inequality holding for all $x,y,z\in \mathbb R$) attains its equality at $x=y=z$ but also at $x:y:z=\sin^2\dfrac{4\pi}{7}:\sin^2\dfrac{2\pi}{7}:\sin^2\dfrac{\pi}{7}$, an equality condition invisible when one restricts oneself to the dense subset of rationals.
Edit. Please look at the comments by Todd:
First note that,
However, as Todd has pointed out, the gap in the argument is that: the above points (1-2; I haven't thought about 3 yet) do not suffice to conclude that $LX^T$ has full rank. Since the conclusion is true (using Noam's or Darij's proofs), maybe there is a way to "rescue" the proof outline below---and if not, then I'll still let this "answer" hang in here to show an example of "what type of proof does not work for this problem!"
Here is an attempt at a Vandermonde-based proof.
This proof below is an adaptation of (the proof of) Theorem 4.3.3 from this book by Bapat and Raghavan (their result is cast in terms of positive definite matrices).
Let $A=\lambda x^T$ (where $\lambda=(\lambda_1,\ldots,\lambda_n)$; likewise $x=(x_1,\ldots,x_n)$. and consider the Schur matrix $[e^{a_{ij}}]$. By direct expansion we have
\begin{equation*} [e^{a_{ij}}] = I + A + \frac{A^{(2)}}{2!} + \ldots + \frac{A^{(k)}}{k!} + \ldots \end{equation*} where $A^{(k)}$ is the Schur power of matrix $A$. We see that, \begin{equation*} A^{(2)} = (\lambda x^T) \circ (\lambda x^T) = (\lambda \circ \lambda)(x \circ x)^T. \end{equation*} Inductively, we obtain that $A^{(k)} = \lambda^{\circ (k)}x^{\circ (k)^T}$ (Schur powers), for $k=1,2,\ldots$.
Thus, it follows that \begin{equation*} [e^{a_{ij}}] = LX^T, \end{equation*} where $L$ and $X$ are infinite matrices with columns given by \begin{eqnarray*} L &=& \begin{pmatrix} \mathbf{1}, \lambda, \frac{\lambda^{\circ (2)}}{\sqrt{2!}},\ldots,\frac{\lambda^{\circ (k)}}{\sqrt{k!}},\ldots \end{pmatrix}\\\\ X &=& \begin{pmatrix} \mathbf{1}, x, \frac{x^{\circ (2)}}{\sqrt{2!}},\ldots,\frac{x^{\circ (k)}}{\sqrt{k!}},\ldots \end{pmatrix}, \end{eqnarray*} where $\mathbf{1}$ denotes the vector of all ones.
The desired invertibility of $[e^{a_{ij}}]$ will follow if we show that each of the matrices $L$ and $X$ has $n$ linearly independent columns. Since the $\lambda_i$ are distinct (given the ordering), as are the $x_i$, as per assumption, the Vandermonde matrix \begin{equation*} V = \begin{pmatrix} 1 & x_1 & x_1^2 & \cdots & x_1^{n-1}\\\\ 1 & x_2 & x_2^2 & \cdots & x_2^{n-1}\\\\ \vdots & \vdots & \vdots & \vdots\\\\ 1 & x_n & x_n^2 & \cdots & x_n^{n-1} \end{pmatrix} \end{equation*} is nonsingular, which shows already that the first $n$ columns of $X$ are linearly independent. A similar argument applies to $L$. Thus, their product $LX^T$ also has full rank, and its determinant is nonzero as desired.
