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I've been struggling with the following exercise for quite some time already:

Consider a linear space $\mathbb{V} = \mathcal{C}\left(\left[a, b\right]\right)$ and let $f_{1},\ldots, f_{n}$ be linearly independent functions in $\mathbb{V}$. Prove there exist numbers $a \leq x_{1} < \cdots < x_{n} \leq b$ such that $$ \det \begin{bmatrix} f_{1}(x_{1}) & f_{1}(x_{2}) & \cdots & f_{1}(x_{n})\\ f_{2}(x_{1}) & f_{2}(x_{2}) & \cdots & f_{2}(x_{n}) \\ \vdots & \vdots & \ddots & \vdots \\ f_{n}(x_{1}) & f_{n}(x_{2}) & \cdots & f_{n}(x_{n}) \end{bmatrix} \neq 0.$$

The statement is extremely easy to prove by means of induction. However, I'm interested if there's another (and more elegant) proof which doesn't involve induction.

Any hints appreciated.

J. W. Tanner
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arystarch
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2 Answers2

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Proceed by contrapositive. We suppose that for all $a\leq x_1 < \cdots < x_n \leq b$, $$ \det \begin{bmatrix} f_{1}(x_{1}) & f_{1}(x_{2}) & \cdots & f_{1}(x_{n})\\ f_{2}(x_{1}) & f_{2}(x_{2}) & \cdots & f_{2}(x_{n}) \\ \vdots & \vdots & \ddots & \vdots \\ f_{n}(x_{1}) & f_{n}(x_{2}) & \cdots & f_{n}(x_{n}) \end{bmatrix} = 0. $$ Equivalently, the above holds for all choices of $x_1,\dots,x_n \in [a,b]$. Consider the subspace of $\Bbb R^n$ defined by $$ U = \operatorname{span}\{(f_1(x),\dots,f_n(x)) : x \in [a,b]\}. $$ Suppose for the purpose of contradiction that $U = \Bbb R^n$. It follows that there exist vectors $v_1,\dots,v_n \in U$ that span $\Bbb R^n$. If we take these vectors as the columns of a matrix, then we end up with an $n \times n$ matrix of the form above; this matrix has linearly independent columns, which means that its determinant is non-zero. This contradicts our assumption.

So, $U$ is necessarily a proper subspace of $\Bbb R^n$. Select any non-zero $c = (c_1,\dots,c_n) \in U^\perp$. By definition, we have $c^Tv = 0$ for all $v \in U$. That is, for every $x \in [a,b]$ we have $$ c_1 f_1(x) + \cdots + c_n f_n(x) = 0. $$ That is, the functions $f_1,\dots,f_n$ are linearly dependent.

The conclusion follows.

The proof by induction, since I was curious. Reduce from the $n$-case to the $(n-1)$-case by noting that

$$ \det \pmatrix{ f_{1}(x_{1}) & f_{1}(x_{2}) & \cdots & f_{1}(x_{n})\\ f_{2}(x_{1}) & f_{2}(x_{2}) & \cdots & f_{2}(x_{n}) \\ \vdots & \vdots & \ddots & \vdots \\ f_{n}(x_{1}) & f_{n}(x_{2}) & \cdots & f_{n}(x_{n}) } = \\ \det\pmatrix{ f_{1}(x_{1}) & f_{1}(x_{2}) & \cdots & f_{1}(x_{n})\\ 0 & f_{2}(x_{2}) - \frac{f_2(x_1)}{f_1(x_1)} f_1(x_2) & \cdots & f_{2}(x_{n}) - \frac{f_2(x_1)}{f_1(x_1)}f_1(x_n) \\ \vdots & \vdots & \ddots & \vdots \\ 0 & f_{n}(x_{2}) - \frac{f_n(x_1)}{f_1(x_1)} f_1(x_2) & \cdots & f_{n}(x_{n}) - \frac{f_n(x_1)}{f_1(x_1)}f_1(x_n) } = \\ f_1(x_1) \det\pmatrix{ f_{2}(x_{2}) - \frac{f_2(x_1)}{f_1(x_1)} f_1(x_2) & \cdots & f_{2}(x_{n}) - \frac{f_2(x_1)}{f_1(x_1)}f_1(x_n) \\ \vdots & \ddots & \vdots \\ f_{n}(x_{2}) - \frac{f_n(x_1)}{f_1(x_1)} f_1(x_2) & \cdots & f_{n}(x_{n}) - \frac{f_n(x_1)}{f_1(x_1)}f_1(x_n) } $$ and defining $g_j(x) = f_{j+1}(x) - \frac{f_{j+1}(x_1)}{f_1(x_1)}f_1(x)$ for $j = 1,\dots,n-1$.

Ben Grossmann
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(I'll call this a recursive algorithm rather than mathematical induction, but one may disagree.)

Let $\mathbf f=(f_1,f_2,\ldots,f_n)^T$.

  • Pick any nonzero vector $v_1$.
  • Since $f_1,\ldots,f_n$ are linearly independent, there exists some $x_1$ such that $v_1^T\mathbf f(x_1)\ne0$.
  • Pick any nonzero vector $v_2\perp\mathbf f(x_1)$ (that is, $v_2^T\mathbf f(x_1)=0$).
  • Since $f_1,\ldots,f_n$ are linearly independent, there exists some $x_2$ such that $v_2^T\mathbf f(x_2)\ne0$.
  • (Continue in this manner...)
  • Pick any nonzero vector $v_n\perp\{\mathbf f(x_1),\mathbf f(x_2),\ldots,\mathbf f(x_{n-1})\}$.
  • Since $f_1,\ldots,f_n$ are linearly independent, there exists some $x_n$ such that $v_n^T\mathbf f(x_n)\ne0$. Now $$ \pmatrix{v_1^T\\ v_2^T\\ \vdots\\ v_n^T}\pmatrix{\mathbf f(x_1)&\mathbf f(x_2)&\cdots&\mathbf f(x_3)} $$ is an upper triangular matrix with nonzero diagonal entries. Hence it is invertible and $\det\pmatrix{\mathbf f(x_1)&\mathbf f(x_2)&\cdots&\mathbf f(x_3)}\ne0$.

By the way, note that neither the continuity of $\mathbf f$ nor the compactness/connectedness of the domain of $\mathbf f$ are relevant. The above proof works as long as $f_1,\ldots,f_n$ are linearly independent (which implies that their common domain has at least $n$ elements, if that matters).

user1551
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