Are the set of vectors linearly dependent?

Question

Are the set of vectors linearly dependent?

1. $ \{ e^{x}, e^{-x}\} $ in $\mathcal{F} (\mathbb{R} ,\mathbb{R} )$

2. $ \{ \frac{1}{x-1}, \frac{1}{x + 1} \} $ in $ \mathcal{F} (]-1,1[,\mathbb{R})$

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1. For $a,b \in \mathbb{R}$,

$ae^{x} + be^{-x} = 0$

We know that: $e^x > 0 $ for all $x \in \mathbb{R} $

I am not sure if I can past to $a = b = 0$.

2. For $a,b \in \mathbb{R}$

$\frac{a}{x-1} + \frac{b}{x + 1} = 0 \implies x(a+b) + a - b =0$

Which does not tell anything about $a$ and $b$.

Answer 1

Guide:

Try to substitute some value of $x$, for example, in the first example, we can let $x=0$ and we obtain $a+b=0$, try to obtain another condition for $a$ and $b$ by letting $x$ equal to another value and then you can solve for $a$ and $b$.

yes, they are linearly independent.