Show that $\mathrm{span}(v_1,\dots,v_k) = \mathrm{span}(v_1,\dots,v_k,v)$ Show that $\mathrm{span}(v_1,\dots,v_k) = \mathrm{span}(v_1,\dots,v_k,v)$ if and only if $v$ is in $\mathrm{span}(v_1,\dots,v_k)$
I am thinking that if $v$ is in $\mathrm{span}(v_1,\dots,v_k$) it must be one of the elements and can be written $\mathrm{span}(v_1,\dots,v_k,v)$ but I'm not sure how to show this mathematically.

Prove the implication two ways.
First prove the forward direction $\implies $:
Let $V=\text{span}(v_1, \ldots, v_k)$ and $W = \text{span}(v_1, \ldots, v_k, v)$ and assume $V=W$. We will prove that $v \in V$. This is trivial since $v \in W = V$ and thus $v \in V$.

Now we prove the backwards implication $\impliedby$:
Assume $v \in V$. We need to prove that $V = W$. To do this, we pick any arbitrary vector $w \in W$ and prove that it is in $V$ and we pick any arbitrary vector $x \in V$ and prove it is in $W$, essentially proving that $V \subseteq W$ and $W \subseteq V$.
Let $x$ be any vector in $V$. Then $x = a_1v_1 + a_2v_2 + \ldots + a_k v_k$ for some coefficients $a_i$. It is evident that $x \in W$ since we can write $x$ as a linear combination of $W$ as such: $x = a_1v_1 + a_2v_2 + \ldots + a_k v_k + 0v$. Thus $V \subseteq W$.
Now let $w$ be any vector in $W$. Then $w = b_1v_1 + b_2v_2 + \ldots + b_kv_k + bv$ and since $v \in V$, it can be written as a linear combination of $(v_1, \ldots, v_k)$ i.e. $ v = c_1v_1 + c_2v_2 + \ldots c_kv_k$ for some coefficients $c_i$. We can then substitute this into the linear combination of $w$:
$$
\begin{align}
w &= b_1v_1 + b_2v_2 + \ldots + b_kv_k + bv \\
  &= b_1v_1 + b_2v_2 + \ldots + b_kv_k + b(c_1v_1 + c_2v_2 + \ldots c_kv_k) \\
  &= (b_1 + bc_1)v_1 + (b_2 + bc_2)v_2 + \ldots + (b_k + bc_k)v_k 
\end{align}
$$
and thus $w$ can be written as a linear combination of $(v_1, \ldots, v_k)$ and so $w \in V$. Since our choice of $w$ was arbitrary, $W \subseteq V$. 
Since we proved that $V \subseteq W$ and $W \subseteq V$, it follows that $V=W$ as required. $\tag*{$\blacksquare$}$
Note: I would normally use summation notation but pedagogically i've found writing the combinations out explicitly can sometimes be easier to grasp initially for students.
