# Determining whether this equation is linearly independent

linear algebraordinary differential equations

I am given 2 sets of equations and I want to determine if they're linearly independent or not.

1. $$e^{-x}, -2e^{2x}+5e^{-x}$$
2. $$x|x|, x^2$$

The definition of linear independence: if there are no constants $$a, b \not= 0$$ such that $$ax_1(t)+bx_2(t)=0$$. For the first set of equations, I can tell that this is linearly independent because of the term $$-2e^{2x}$$ will never go away. For the second set of equations, I'm not so sure. If $$x \geq 0$$, then this equation is not linearly independent. But $$x \geq 0$$ is not stated as a restriction. I just need someone to hopefully confirm my findings and see if my hypothesis is correct.

Suppose $$ae^{-x}+b(-2e^{2x}+5e^{-x})=0$$. Divide throughout by $$e^{2x}$$ and take limit as $$x \to \infty$$. You get $$b=0$$ from which you can see that $$a$$ must also be $$0$$. Hence, the first see is linearly independent.
Suppose $$ax|x|+bx^{2}=0$$. Put $$x=1$$ to get $$a+b=0$$ and $$x=-1$$ to get $$-a+b=0$$. Use these two equations to show that $$a=b=0$$. So the second set is also linearly independent.