In the book Functional Analysis,Sobolev Spaces and Partial Differential Equations of Haim Brezis we have the following lemma:
Lemma. Let $X$ be a vector space and let $\varphi, \varphi_1, \varphi_2, \ldots, \varphi_k$ be $(k + 1)$ linear functionals on $X$ such that $$ [\varphi_i(v) = 0 \quad \forall\; i = 1, 2, . . . , k] \Rightarrow [\varphi(v) = 0]. $$
Then there exist constants $\lambda_1, \lambda_2, \ldots, \lambda_k\in\mathbb{R}$ such that $\varphi=\lambda_1\varphi_1+\lambda_2\varphi_2+\ldots+\lambda_k\varphi_k$.
In this book, the author used separation theorem to prove this lemma. I would like ask whether we can use only knowledge of linear algebra to prove this lemma.
Thank you for all helping.