Suppose that $(v_1, v_2,\ldots, v_m)$ is an ordered basis of $V$, and $(w_1, w_2,\ldots, w_n)$ is an ordered basis of $W$. If the transform matrix from $V$ to $W$ has a first column of all $0$'s, what are the implications for the basis vectors? All I can think of is that the first element of each $v_i$ in $V$ is irrelevant in the the transform. Can someone give the general idea of what it means?
The original question is that $(v_1, \ldots, v_m)$ is a basis of V. The transform matrix of dimension n x m from V to W has the first column all 0s or a one in the first column first row. Prove that there exists a basis $(w_1,\ldots, w_n)$ of W.