I've been thinking about this for almost a day and I have given up. I just get stuck in an invalid argument and dunno how else to do this.
So the question is:
Let $s>1$ and let $f_1,...,f_s$ be a basis of the ideal $I$ in $k[x_1,...,x_n]$. If $f_i \neq 0$ for all $i$, then show that for any $i$ and $j$, zero can be written as a linear combination of $f_i$ and $f_j$ with nonzero polynomial coefficients.
So essentially, I have to show that for $k_i,k_j$ in $k[x_1...x_n]$, if $k_if_i+k_jf_j=0, k_i,k_j \neq 0$.
I first started with assuming $k_if_i+k_jf_j=0$ and then showing that $k_i,k_j$ are not zero. I then said because $I$ is an ideal, $0$ is in $I$, and thus $k_1f_1+...+k_sf_s = 0$ and tried to use the fact, but that didn't end up proving anything I wanted.
I've been fiddling with it but no, I am starting to think using $k_1f_1+...+k_sf_s = 0$ doesn't help, but I have no idea what else to use otherwise....
Please please help I am so stressed with this, it seemed so simple at first and I think I am missing something basic, I just can't figure what and it is unbelievably irritating :/
I really appreciate your help, thanks so much!