https://i.stack.imgur.com/ydyNi.jpg
In the image linked above, my textbook says that being linearly independent over an interval is equivalent to the wronskian being 0 at an (arbitrary?) time in the interval.
This equivalency is utilized in the following example, where they conclude that since the wronskian is not equal to zero at a point on the interval at time = 1 it is independent over the interval. However, it is zero at time = 0, so how were they able to make that conclusion?
https://i.stack.imgur.com/RZfAy.jpg
The only way I can make sense of this is that the equivalency condition in the first picture means: it is linearly independent on the interval iff the wronskian is not equal to zero on AT LEAST one point in the interval, not necessarily any point.
However, even if I interpreted this to be the condition, then the paragraph at the bottom of the second picture doesn't make sense where they cite an example in which a function is independent over an interval even though the wronskian is ALWAYS zero.
There seems to be contradictions throughout this, at least to me (a beginner math student). Any help would be greatly appreciated, because I'm left very confused.
Thank you!