# Prove that the set of real-valued functions $\{|x - \lambda_1|, |x - \lambda_2|,\ldots, |x - \lambda_n|\}$ is linearly independent

I am looking for another solution to the problem below.

Prove that the set of real-valued functions $$\{|x - \lambda_1|, |x - \lambda_2|,\ldots, |x - \lambda_n|\}$$ is linearly independent over $$\mathbb R$$ for $$\lambda_i \neq \lambda_j, i \neq j$$ ($$1 \le i, j \le n$$).

I have been able to come up with a proof:

• Reorder the set $$\{\lambda_1, \lambda_2, \ldots, \lambda_n\}$$ such that $$\lambda_1 \lt \lambda_2 \lt \cdots \lt \lambda_n$$.
• Differentiate both sides of the equation: $$\alpha_1|x - \lambda_1| + \alpha_2|x - \lambda_2| + \cdots \alpha_n|x - \lambda_n| = 0$$ on intervals $$(\lambda_1, \lambda_2)$$, $$\ldots$$, $$(\lambda_{n-1}, \lambda_n)$$, $$(\lambda_n, +\infty)$$.
• Obtain the system of $$n$$ equations, from which we can conclude $$\alpha_1 = \ldots = \alpha_n = 0$$:$$\left\{ \begin{array}{c} \alpha_1-\alpha_2+\cdots-\alpha_n=0 \\ \alpha_1+\alpha_2+\cdots-\alpha_n=0 \\ \ldots\\ \alpha_1+\alpha_2+\cdots+\alpha_n = 0 \end{array} \right.$$

Are there anything wrong with my proof and can you folks come up with another?

• Your proof is (correct and) nice! Mar 1, 2021 at 5:03

Here's a different proof, also starting with reordering the $$\lambda_j$$ in increasing order. Suppose that $$\alpha_1|x - \lambda_1| + \alpha_2|x - \lambda_2| + \cdots \alpha_n|x - \lambda_n| = 0$$. If $$\alpha_j\ne0$$, then near $$x=\lambda_j$$ we have $$|x - \lambda_j| = \frac1{\alpha_j} \bigg( \sum_{ij} \alpha_i(\lambda_i-x) \bigg);$$ but this is impossible since the right-hand side is a linear function of $$x$$ while the left-hand side is not near $$x=\lambda_j$$.

If $$\alpha_1|x - \lambda_1| + \alpha_2|x - \lambda_2| + \cdots \alpha_n|x - \lambda_n| = 0$$ then $$\alpha_i$$ must be $$0$$ for each $$i$$ simply because RHS is differentiable at $$\lambda_i$$ but LHS is not unless $$\alpha_i=0$$ : Note that all terms except the $$i-$$th one are differentiable at $$\lambda_i$$.

• @BrianMoehring Thank you! Mar 1, 2021 at 6:00