Difficulty proving that if set of vectors v's span doesn't contain another vector b, then the set of vectors v & vector b isn't linearly independent.

Solution 1:

I think the issue might be with the framing of the problem. Let's break it down. First, we have that $\{v_1,...,v_n\}\subset A$ be defined as a generating set. This implies that $\text{span}\{v_1,...,v_n\}=A$. Now we are told that the set of vectors does not contain $b$, so we have that $b\notin\{v_1,...,v_n\}$, but we will have that $b\in A$. Now to see that $\{v_1,...,v_n,b\}$ is linearly dependent it suffices to show that there is a non-trivial linear combination giving $0$. As $\{v_1,...,v_n\}$ is a generating set of $A$, given any $w\in A$ there exist $c_i\in F$ such that $\sum c_iv_i=w$, so as $b\in A$ there exist $c_i\in F$ such that $\sum c_iv_i=b$, then we have that $(\sum c_iv_i)-b=0$ is a non-trivial linear combination of zero from the set $\{v_1,...,v_n,b\}$, so we have that this is a linearly dependent set.