linear independence of vector functions
I am given the question: Let I be an interval and let y1, . . . , yM be vector functions I - C^N. Show that if t0 € I and y1(t), . . . , yM(t) € C^N are linearly independent vectors then y1,....,ym are linearly independent functions.
Argue by contradiction. Thus assume the
M functions y1(t), . . . , yM(t) are linearly dependent i.e.that there exist constants c1, . . . , cM not all zero such that
cjyj(t) = 0 for all t € I. I want to evaluate at t=t0 but I'm not sure what to do next.
Any suggestions would be awesome :)
Re: linear independence of vector functions
if the functions are linearly independent, then there exists constants
such that for EVERY t in I (that is, the sum is the constant function which has the
0-vector of as it's value for every t).
but this means that for any t in I, is a linearly dependent set of vectors, the desired contradiction.