linear independence in $F(\mathbb R,\mathbb R)$

Question

For each subset of each vector space, justify whether their vectors are linearly independent or dependent:

$$\{f,g,h\}, f(x)=e^{2x} , g(x)=x^2 , h(x)=x,\mbox{ in }F(\mathbb R,\mathbb R).$$

Answer 1

Assuming you meant $f(x) = e^{2x}, g(x) = x^2, h(x) = x$, you need to show that if there are constants $\alpha_f, \alpha_g, \alpha_h$ such that $\alpha_f f + \alpha_g g + \alpha_h h = 0$ (that is, equal to the the function $t \mapsto 0$), then these constants must be zero.

Some tricks for this sort of problem are evaluating $\alpha_f f + \alpha_g g + \alpha_h h$ at various points, or differentiating an appropriate number of times and evaluating.

Let $\phi(x) = \alpha_f f(x) + \alpha_g g(x) + \alpha_h h(x)$. We have $\phi(x) = 0$ for all $x$. So, $\phi(0) = \alpha_f = 0$. So we know that $\alpha_f = 0$. We note that since $\phi$ is contstant, $\phi' = 0$, and so $\phi'(0) = \alpha_g g'(0) + \alpha_h h'(0) = \alpha_h = 0$. So we know that $\alpha_h = 0$. Finally, since $\phi(1) = \alpha_g g(1) = \alpha_g = 0$, we have $\alpha_g = 0$.

Hence $f,g,h$ are linearly independent.