0
$\begingroup$

Let's say $~a,~ b~$ are linearly independent functions on an interval $~I~$. Are they linearly independent on any interval $~J~$ contained in $~I~$? If so, how do I prove it?

Let's say $~a,~ b~$ are instead linearly dependent functions on an interval $~I~$. Are they linearly dependent on any interval $~J~$ contained in $~I~$? If so, how do I prove it?

I have a feeling I'm supposed to use the Wronskian determinant for these but I'm not sure how to apply it.

$\endgroup$
  • 1
    $\begingroup$ What happens for $a= x$, $b= x \mathbf{1}_{(-\infty,1]}+ x^2 \mathbf{1}_{(1,\infty)}$ ? $\endgroup$ – Sungjin Kim Oct 20 '16 at 23:53
  • 1
    $\begingroup$ No, don't use the Wronskian determinant! Use your brain. Look for counterexamples. As a small hint, the answer to one of those questions is Yes, and the answer to the other one is No. $\endgroup$ – TonyK Oct 21 '16 at 0:12
0
$\begingroup$

False:

Take $~f(x) = x^2~$ and $~g(x) = x|x|~$. Then $~f, ~g~$ linearly independent over $~[−1, 1]~$ but dependent over $~[0, 1]~$.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.