My question is deceptively simple. Let $v_1, \dots, v_m \in \mathbb{R}^n$ be a set of vectors linearly independent. If we multiply them by a matrix A, such that the vectors $Av_1, \dots, Av_m$ are linearly dependent, which of the following sentences is true and why or why not.
- rank($A$) < m.
- rank($A$) < n.
I am really stuck with this. I've made some calculations like, if $Av_1, \dots, Av_m$ are linearly dependent, then
$\lambda_1(Av_1) + \dots + \lambda_m(Av_m) = 0$
has more than one solution. Rewriting the last expression i concluded that
$A(\lambda_1 v_1 + \dots + \lambda_m v_m) = 0$
And here I got stuck.