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?  

The standard notation for the vector space of all functions $\Bbb R^+\to\Bbb R$ is $\Bbb R^{\Bbb R^+}$. This is sort of awkward and I've also seen the notation $\mathcal F(0,\infty)$ used. As long as you define your notation properly I don't think notation matters much here.
As for showing that $f(x)$ and $g(x)$ are linearly independent in $\mathcal F(0,\infty)$, we can use the definition. Seeking a contradiction, suppose that there exist scalars $a$ and $b$ not both zero such that
$$
af(x)+bg(x)=0\tag{1}
$$
for all $x>0$. Then 
$$
ax=-\frac{b}{x}\tag{2}
$$
for all $x>0$. In particular, plugging $x=1$ and $x=2$ into (2) gives
$$
a=-b\tag{3}
$$
and
$$
2a=-\frac{b}{2}\tag{4}
$$
Plugging (3) into (4) gives
$$
2a=\frac{a}{2}
$$
which implies $a=0$. But then (3) implies $b=0$ too, a contradiction. Hence $f(x)$ and $g(x)$ are linearly independent in $\mathcal F(0,\infty)$.
The key point here is that the equation (1) holds for all $x>0$ which is what allows us to arrive at our desired contradiction. The reason (1) is assumed to hold for all $x>0$ is because the elements of $\mathcal F(0,\infty)$ are functions which are equal if and only if they are equal at every point of $\Bbb R^+$.
