In a vector space $V$ over $\mathbb{F}$, assume $\alpha _{1} , \ldots , \alpha _{n}$ are linearly independent but that $\beta , \alpha _{1} , \ldots , \alpha _{n}$ are linearly dependent for each $\beta \in V$. Prove that $\alpha_1 , \ldots , \alpha_{n}$ forms a basis of $V$.
Here's my attempt:
If $\alpha_1 , \ldots , \alpha_{n}$ forms a basis of $V$, then we are done. If not, let $\beta$ be any element in $V \setminus \text{Sp} \{ \alpha_1 , \ldots , \alpha_{n} \}$. Then $\alpha_1 , \ldots , \alpha_{n}, \beta$ would be linearly independent by the linear independence lemma. This contradicts our assumption. Thus, $\alpha_1 , \ldots , \alpha_{n}$ forms a basis of $V$.
Is this proof okay? This is a problem frm Katsumi Nomizu's Fundamentals of Linear Algebra.