- #1
srfriggen
- 307
- 6
This is from my text, "Linear Algebra" by Serge Lang, pg 11:
-The two functions et, e2t are linearly independent. To prove this, suppose that there are numbers a, b such that:
aet + be2t=0
(for all values of t). Differentiate this relation. We obtain
aet + 2be2t = 0.
Subtract the first from the second relation. We obtain be2t=0, and hence b=0. From teh first relation, it follows that aet=0, and hence a=0. Hence et, e2t 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?
-The two functions et, e2t are linearly independent. To prove this, suppose that there are numbers a, b such that:
aet + be2t=0
(for all values of t). Differentiate this relation. We obtain
aet + 2be2t = 0.
Subtract the first from the second relation. We obtain be2t=0, and hence b=0. From teh first relation, it follows that aet=0, and hence a=0. Hence et, e2t 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?