I'm reading deBoor's (wonderful) book "A practical guide to splines", revised edition. I'm doing some of the exercises at the end of each chapter just to fix the main ideas before going ahead...
Let's go to the question: Exercise 4 of chapter VII, point (c) claims that the linear independence of $n$ functions $\varphi_i$ belonging to a finite-dimensional linear space (quoted) "is invariably shown by exhibiting a corresponding sequence $\lambda_1,\ldots,\lambda_n$ of linear functionals for which the matrix $(\lambda_i\varphi_j:i,j=1,\ldots,n)$ is obviously invertible, that is, is triangular with non-zero diagonal elements."
In general why is this so? I never met such an argument in my one elementary linear algebra course...