Can this be a valid complete answer for this question?

Question

Assume $f_1,f_2,f_3$ are linearly independent functions.Show that the addition to the set of one of these functions say, $f_1$ makes the new set linearly dependent.
I show that $(1)f_1+(-1)f_1+(0)f_2+(0)f_3=0$.
Is it enough? Is that all that is asked here?

Answer 1

This is incorrect. The set $\{f_1,f_2,f_3\}\cup \{f_1\}$ is just $\{f_1,f_2,f_3\}$ again and so still linearly independent. That is one reason why I am always extremely unhappy when people introduce the concept of linear dependence in terms of sets of vectors. One should rather use families of vectors.

If we silently interpret each occurence of "set" as "family" in the problem statement, then everything is correct.