Linear independence of the vector in $\mathbb R^{\mathbb R}$

Question

Try to conclude is vectors $v,u,w$ linear independent in vector space $\mathbb R^{\mathbb R}$ where $v(x)=|x-2|, u(x)=|x-3|, w(x)=|x-5|$

I know that I need to use some scalar and show if they are zero or not, but here I am confuse, I need to take some scalar from $\mathbb R$, or not? Because i know that $\mathbb R^{\mathbb R}=\{f\colon \mathbb R\to \mathbb R\}$, can someone help me?

Answer 1

You have to show that if $a|x-2|+b|x-3|+c|x-5|=0$ for all real numbers $x$ then $a=b=c=0$. Just put $x=2$, then $x=3$ and then $x=5$. You will get 3 equations in $a,b,c$ . From these equations it is fairly easy to show that $a=b=c=0$.

Answer 2

The space of functions from $\mathbb{R}$ to $\mathbb{R}$ is a vector space over $\mathbb{R}$, so you need to take scalars in $\mathbb{R}$ to check linear independence.

Answer 3

Yes, scalars from $\mathbb R$. Let $a,b,c \in \mathbb R$ such that

$$au(x)+bv(x)+cw(x)=0$$

for all $x \in \mathbb R$. You have to show that $a=b=c=0$.