-1
$\begingroup$

The question is: Suppose that f and g are linearly independent functions on the real line. Prove that there exists $t_0\in \mathbb{R}$ such that $W(f,g)(t_0) \neq 0$. This seems rather simple, but I am having trouble putting together a solid proof. Any suggestions for a starting point?

I know that linear independence means the matrix of f and g will have a non-zero determinate, but how do I apply this to the Wronskian?

2 Answers 2

0

Hint: If $W(f,g)(t)=0$ for all $t\in\mathbb{R}$, this means the matrix $$A(t)=\begin{bmatrix}f(t) & g(t)\\ f'(t) &g'(t)\end{bmatrix}$$ is singular for every $t$. Then what can you say about the columns of $A(t)$?

  • 0
    You could say the columns are linearly dependent, correct? @Frank Lu – 2017-02-13
  • 0
    So this proof could work by contradiction? @Frank Lu – 2017-02-13
  • 0
    That's the point. – 2017-02-13
  • 0
    There's a distinction between contradiction and contrapositive. This example falls in the latter category. – 2017-02-13
-1

If $\forall t\in \mathbb{R}$ such that $W(f,g)(t) = 0$ so $$W(f,g)=‎\left|\begin{array}{rr}f&g\\f'&g'\end{array}\right|=0$$ so ‎$$\dfrac{f'}{f}=\dfrac{g'}{g}$$ shows $$\ln f=\ln Cg$$ for a constant $C$, this means $f=Cg$.