Let $k$ be an infinite field. Assume that $n,m$ are two positive integers such that $n>m$. Consider symmetric matrices $A_1,\dots,A_m$ of size $n\times n$. Suppose for each $i=1,\dots,m$, every column of the matrix $A_i$ has a non-zero entry which is not on the main diagonal. Is the following implication correct?

If, for every vector $v \in k^n$ the vectors $A_1v,\dots,A_mv$ are linearly dependent, then the matrices $A_1,\dots,A_m$ are linearly dependent.