I am reading through Friedberg, Spence, and Insel's book on Linear Algebra. In section 1.5, the section introducing the notions of linear dependence and independence, the following statement is made:
Given a vector space $V$, if no proper subset of $S$ of $V$ generates the span of $S$, then $S$ is linearly independent.
I understand this statement, and I am able to prove it.
Then, almost immediately afterwards, the following theorem is given:
Let $S$ be a linearly independent subset of $V$, and let $v$ be a vector in $V$ which is not in $S$. Then $S\cup \{ v \}$ is linearly dependent if and only if $v\in\text{span}(S)$.
I also understand and am able to prove this statement.
However, in between these two statements, a claim is made that they are equivalent. And this is where I am having trouble. I am unable to see how these two theorems mean the same thing. Any help clarifying why these two are equivalent would be greatly appreciated.
Note that the definition of linear independence in this text is as follows:
A subset of a vector space that is not linearly dependent is called linearly independent.
and the definition of linear dependence used here is:
A subset $S$ of a vector space $V$ is called linearly dependent if there exist a finite number of distinct vectors $u_1, u_2, ..., u_n$ in $S$ and scalars $c_1, c_2, ..., c_n$, not all zero, such that $\sum_{i=0}^n c_iu_i = 0$.