For several functions, does dividing by a common term (e.g. x) affect linear independence of the functions?

Question

I am a student taking a course in linear algebra and differential equations. The textbook question asks to verify that $e^{4x}$, $xe^{4x}$, and $x^{2}e^{4x}$ are linearly independent on $(-\infty,\infty ) $. Of course, this can be verified using a Wronskian of the given functions, but I was wondering whether it was mathematically valid to divide each function by the common term of $e^{4x}$, giving me 1, $x$, and $x^{2}$, which are much easier to compute the Wronskian for, and by extension, whether in general functions can be "simplified" in this way to make verification of linear independence easier? If possible, some proof or mathematical explanation as to why or why not this can be done would also be appreciated.

Answer 1

Nice question! Yes, it can be done. Suppose that the functions $f_1,\ldots,f_n$ are linearly dependent. Then there are numbers $a_1,\ldots,a_n$, not all of which are $0$, such that $a_1f_1+\cdots+a_nf_n=0$. But then, for any function $g$, $a_1gf_1+\cdots+a_ngf_n=g\times0=0$ and therefore the functions $gf_1,\ldots,gf_n$ are linearly dependent. And, assuming that $g$ has no zeros, you can do the same in the other direction, dividing by $g$. That is, if $gf_1,\ldots,gf_n$ are linearly dependent, then $f_1,\ldots,f_n$ are linearly dependent too.