How can the function $\phi_1 (x) = x^2 \quad \& \quad \phi_2(x) = x |x|$ be linearly independent ?

Question

In the book of Introduction to Ordinary Differential Equations by Coddington , at page 64 in question 3, it is asked to

show that the functions $$\phi_1 (x) = x^2 \quad \& \quad \phi_2(x) = x |x|$$ are linearly independent on $- \infty < x < \infty$.

However, I do not understand how can this be ?

I mean if we restrict ourselves to positive and negative values of $x$, the function $\phi_2$ becomes $$\phi_2(x) = \pm x^2,$$ respectively, hence in either case, they are linearly dependent, so what am I missing in here ?

Answer 1

The keyword is for all $x\in \mathbb{R}.$

Suppose $\forall x \in \mathbb{R}$,$$c_1 \phi_1(x)+c_2\phi_2(x)=0$$

$$c_1x^2+c_2|x|x=0$$

If $x=1$, we have $c_1+c_2=0$.

If $x=-1$, we have $c_1-c_2=0$.

Hence we have $c_1=c_2=0$.