The dependence between vectors and the dependence between matrices

Question

Let $P_1,\cdots, P_r\in M_{n\times n}$ be matrices, where $r\leq n$. Suppose that $P_1a,\cdots, P_ra$ are not linear independent for any $a\in\mathbb{C}^n$. Could we know that $P_1,\cdots, P_r$ are not linear independent?

Answer 1

$P_1=\pmatrix{1&0\cr0&0\cr}$, $P_2=\pmatrix{0&1\cr0&0\cr}$ are linearly independent. $P_1(a,b)=(a,0)$, $P_2(a,b)=(b,0)$ are never linearly independent.