Thread: linear independence of vector functions

1. linear independence of vector functions

Hi all,

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.

My attempt:

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

$\displaystyle \sum$ 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

2. Re: linear independence of vector functions

if the functions $\displaystyle \{y_1,\dots,y_m\}$ are linearly independent, then there exists constants $\displaystyle \{c_1,\dots,c_m\}$

such that $\displaystyle \sum_{i=1}^m c_iy_i(t) = (0,\dots,0)$ for EVERY t in I (that is, the sum is the constant function which has the

0-vector of $\displaystyle \mathbb{C}^n$ as it's value for every t).

but this means that for any t in I, $\displaystyle \{y_1(t),\dots,y_m(t)\}$ is a linearly dependent set of vectors, the desired contradiction.