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Consider the vector space $V$ of all continues functions on $R$ over the field $R$.

Let $S= \{ \vert x \vert, \vert x-1 \vert, \vert x-2 \vert \}$.

Is the set linearly dependent or independent and span the vector space?

$|x|$ is either $x$ or $-x$ i.e. dependent implying that the whole set is dependent.
But I'm not sure about it.

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1 Answers 1

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Can we achieve $|x|=a|x-1|+b|x-2|$ ?

If yes, the equality must hold for $x=1,2$, giving

$$b=1,a=2.$$

But then, with $x=0$ we get

$$a+2b=0,$$ a contradiction.