Linear dependence and linear independence of functions in linear algebra

Question

I am trying to understand linear dependence and linear independence of real valued functions on a set. Say S. I want to know that using wronskain how can we say that a set S of functions is linearly dependent. I was thinking that if wronskain is zero everywhere on the domain then S is linearly dependent and if at least at one point of the domain wronskain is nonzero then S is linearly independent. Where is the problem actually ?

Answer 1

Let us see the relation between the Wronskain and the linearly dependent and independent solutions. It may be clear your confusion.

Let $f$ and $g$ be two real valued differentiable functions on a set $S = [a,b]$ (say). If Wronskian $W(f,g)(t_{0})$ is nonzero for some $t_{0}$ in $[a,b]$, then $f$ and $g$ are linearly independent on $[a,b]$.

If $f$ and $g$ are linearly dependent then the Wronskian $W(f,g)(t_{0})$ is zero for all $t_{0}$ in [a,b] .

You can also verify this in

1. "Differential Equations" by Shepley L. Ross ($3^{rd}$ Edition, Page 112, Theorem $4.4$).
2. "An Introduction to Ordinary Differential Equations" by Ravi P. Agarwal, Donal O'Regan (Page $118$ (Theorem $17.1$))