Some property of linearly independent linear maps.

Question

let $ f_1, f_2,... ,f_k$ be independent linear maps from a linear space $V$ to $\mathbb{R}$.

Can we say that there exists a vector $v \in V$ such that $f_1(v)=0, f_2(v)=0,... ,f_{k-1}(v)=0$ but $f_k(v) \ne 0$ ?

Answer 1

Yes, we can. Since $f_1$ and $f_2$ are linearly independent, $\ker f_1\neq\ker f_2$. Therefore $\ker(f_1)\cap\ker(f_2)$ has codimension $2$. In particular$$\operatorname{codim}\bigcap_{j=1}^{k-1}\ker f_j>1.$$But $\operatorname{codim}\ker f_k=1$ and therefore$$\bigcap_{j=1}^{k-1}\ker f_j\varsubsetneq\ker f_k.$$In other words, there's a vector $v\in\bigcap_{j=1}^{k-1}\ker f_j$ such that $f_k(v)\neq0$.