Linear independence (functions)

Given a set of functions

${\vec {x}}^{(1)},\ldots ,{\vec {x}}^{(k)}$

Defined in a < t < b. They are linearly independent of the only solution to

$c_{1}{\vec {x}}^{(1)}+\ldots +c_{k}{\vec {x}}^{(k)}=0$

is all c's being 0.

Similar to the Wronskian, as long as it is independent when evaluated at one t, it is sufficient to say that the set is independent.