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.