Question about linear (in)dependency I have a question about linear dependency.
Suppose we have a set $S$ of functions defined on $\mathbb{R}$.
$S = \{e^x, x^2\}$. It seems very intuitive that this set is linear independent. But, we did something in class I'm unsure about.
Proof:
Let $\alpha, \beta \in \mathbb{R}$.
Suppose $\alpha e^x + \beta x^2 = 0$
We need to show that $\alpha = \beta = 0$ is the only option to make sure that this linear combination equals 0.
(Here comes the part I'm unsure about)
Let $x = 0$, then $\alpha e^0 + \beta 0^2 = 0$
$\Rightarrow \alpha  = 0$
But if $\alpha  = 0$ then follows that $\beta = 0$.
So $S$ is linear independent.
My actual question:
Why can we conclude that the set is linear independent, just by saying that $x = 0$ makes it work? Shouldn't we show that it works for all $x \in \mathbb{R}$?
Can someone give a detailed explanation, as I didn't quite understand the teacher's explanation.
Thanks in advance.
 A: We have $$\forall x \in \mathbb{R}, \alpha e^x + \beta x^2 = 0$$
Your goal is to determine all the possible values of $\alpha$ and $\beta$. Clearly, $\alpha=\beta=0$ is a solution, but can there be other solutions?
What are the properties that $\alpha$ and $\beta$ must satisfy?
In particular, we can derive some conditions by fixing certain values of $x$.
By setting $x=0$, we conclude that $\alpha$ must be $0$. We can recover $\beta$ by letting $x=1$.
Clearly, we can get more conditions by fixing more values of $x$, but just from these two conditions, we have concluded that the unique solution is $\alpha=\beta=0$.
A: The two functions are linearly dependent if exists $(\alpha, \beta)$ constants , different from $(0,0)$, such that $\alpha e^x + \beta x^2=0$ for all the $x$  in the considered domain. 
So if for some specific values of $x$ the condition above does not subsists, it does not hold for all $x$, so the functions are independent.
Your teacher attempted and find such a point at $x=0$, for which $(0,\beta)$ results. This solution however does not hold for ,e.g., $x=1$. For those two points no $(\alpha,\beta) \ne (0,0)$ can be found, so the two functions are linearly independent.
