Let $S$ be a spanning set of a vector $V$. If another vector of $V$ is added to $S$, can the new set still be spanning set of $V$?

Question

Let $S$ be a spanning set of a vector $V$. If another vector of $V$ is added to $S$, can the new set still be spanning set of $V$? Justify your answers.

$\underline{Attempt}$

I think that is correct,

Let $S=\{ v_1,v_2,..,v_n \}$ is spanning set of vector space $V$

If another vector$(v_k)$ in $V$ added to the $S$,then $\bar{S}=\{ v_1,v_2,..,v_n,v_k \}$

since $S$ spanning set of vector space $V$

$v_k=\alpha_1v_1+\alpha_2v_2+...+\alpha_nv_n$ where $\{\alpha_1,\alpha_2 ...\alpha_n \}\in R$

so $\bar{S}=\{ v_1,v_2,..,v_n,\alpha_1v_1+\alpha_2v_2+...+\alpha_nv_n \} \subset \{ v_1,v_2,..,v_n \}$

thus new set still be spanning set of $V$.

Is it correct if not please give some hints?

Answer 1

I will write $v_0$ for the new vector instead of $v_k$.

You did not prove what you are supposed to prove. You have to prove that any vector $v \in V$ is a l.c. of vectors from $\overline S$. Since we can write $v=\sum\limits_{i=1}^{n} c_iv_i$ we can write $v=\sum\limits_{i=1}^{n} c_iv_i +0(v_0)$ and that finishes the proof.