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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##

(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}##?

Thanks in advance.

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# I Linear (in)dependency

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