Does $f1(x)f2(y)−f2(x)f1(y)=0\ (\forall x,y\in \mathbb{R})$ mean that vectors $f_1,f_2$ are linearly dependent?

Question

Vector space $V=\mathbb{R}^{\mathbb{R}}$ and $f_1, f_2\in V$. Does $f1(x)f2(y)−f2(x)f1(y)=0\ (\forall x,y\in \mathbb{R})$ mean that vectors $f_1,f_2$ are linearly dependent?

Answer 1

The answer is yes.

First, if $f_1 = 0$, then obviously they're linearly dependent. On the other hand, if there is an $x$ such that $f_1(x) \neq 0$, then for that fixed $x$, we have $$ f_2(y) = \frac{f_2(x)}{f_1(x)} f_1(y) \qquad \forall y \in \Bbb R $$ That is, we have $f_2 = \frac{f_2(x)}{f_1(x)} f_1$, which means that the functions are indeed linearly dependent.