Linear Independence of two functions and differentiation

[srfriggen]

This is from my text, "Linear Algebra" by Serge Lang, pg 11:

-The two functions e^t, e^2t are linearly independent. To prove this, suppose that there are numbers a, b such that:

ae^t + be^2t=0

(for all values of t). Differentiate this relation. We obtain

ae^t + 2be^2t = 0.

Subtract the first from the second relation. We obtain be^2t=0, and hence b=0. From the first relation, it follows that ae^t=0, and hence a=0. Hence e^t, e^2t are linearly independent.



I'm confused as to the "Differentiate this relation". I see it creates a system of equations which can then be used to solve for linear independence, but why does it work?

 

[oli4]

Hi srfriggen

The first statement says that the function of t ae^t+b^2t=0
This must be true for all t, so this function of t is constant (always=0)
therefore, if you look at its derivative, it must always be 0 too.
So the second equation comes out, and since both equations are true, you can put them together and the answer comes out