# If $a,b$ are linearly independent functions on an interval $I$, are they linearly independent on any interval $J$ contained in $I$?

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.

• What happens for $a= x$, $b= x \mathbf{1}_{(-\infty,1]}+ x^2 \mathbf{1}_{(1,\infty)}$ ? – Sungjin Kim Oct 20 '16 at 23:53
• 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. – TonyK Oct 21 '16 at 0:12

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