Just to clarify what was said in class today, I said to rewrite the equation
c1*u + c2*v = 0
a*(something_1) + b*(something_2)=0
Since something_1 and something_2 are both functions of t (a variable), you can pick any t_0 (a constant) and plug it for t so that something_1 and something_2 become constants. I mentioned that you should make a smart choice for t_0 so algebraically the problem becomes simpler. The important thing is that you can pick ANY VALUE for t_0. After making some choice for t_0, if you find that c1=c2=0, then it follows that u and v are linearly independent.