Linear independence of two functions, how to solve I have a problem.  I need to show that $\{f,g\}$ is a linearly independent set in the vector space of all functions from $\mathbb{R}^{+}$ to $\mathbb{R}$, where
$$f(x)=x$$
$$g(x)=\frac1{x}$$
First (and least important), is there a standard notation for this vector space?
Second, I didn't know how to answer this question, so I cheated and checked the solution manual.  There, it basically showed that letting $x=1$ first and $x=2$ second and setting $h(x)=af(x)+bg(x)=0$ gives a system of equation.  Solving shows $a=b=0$, and thus linearly independent.  
Not having any exposure really with this particular vector space, why are we able to pick values for $x$?  We don't do that for $\mathbb{R}^n$ or $\mathbb{F}_n[x]$.  I guess I'm just not accustomed to solving problems in this vector space and am unsure how to procede and why this is the method?  Can someone explain to me why this is the correct way?  

Your statement is that for two constants $a$ and $b$ the function $h(x) = af(x)+bg(x)$ is zero for all values of $x \in \mathbb{R}^+$.
One way to prove independence in cases like this it is taking the derivative of $h(x)$ and assuming $h(x) \equiv 0$. This leads to
$$a - \frac{b}{x^2} = 0$$ for all values of $x$ what is clearly false.
