A family of $n$ non-zero vectors of an $(n-1)$-dimensional vector space must be linearly dependent

Question

I was bored earlier and began to think of the pigeonhole principle, and it came to me that it could be used to show that a family of $n$ non-zero vectors of an $(n-1)$-dimensional vector space must be linearly dependent... but I quickly lost mental track of how I came to that conclusion. I don't need rigorous proof and it's not for any homework problems, I'm just hoping someone might be able to contribute some insight I seem to now have lost.

Answer 1

Really the contradiction argument in Chris Wong's comment is the best way to solve this. If you'd prefer a direct proof, however, here it is.

Any $n-1$ dimensional vector space is spanned by $n-1$ vectors. Let's call them $v_1,\ldots, v_{n-1}$. Now, suppose that you have your family of $n$ non-zero vectors, $w_1,\ldots, w_{n-1}$. Each may be written as a linear combination of the $n-1$ $v_i$ vectors, so we may perform gaussian elimination on $w_1,\ldots, w_{n-1}$. This must at least reduce to one $w_i$ which can be written as a sum of the other $w_j$, which implies the conclusion.