Characterizing linear independence in terms of span

[psie]

TL;DR

I'm reading Linear Algebra by Friedberg, Insel and Spence. Prior to a theorem, they make a statement about linear independence and they claim this can be deduced from the theorem.

Throughout, let $\mathsf V$ be a vector space (the concept of dimension has not been introduced yet). The statement that precedes the theorem below is that if no proper subset of $T\subset \mathsf V$ generates the span of $T$ (where, if I'm not mistaken, $T$ consists of two or more vectors) then $T$ must be linearly independent. Taking the contrapositive,

Claim: If $T\subset\mathsf V$ is linearly dependent, then there is some proper subset $S\subset T$ such that $\operatorname{span}(S)=\operatorname{span}(T)$.

Theorem: Let $S\subset \mathsf V$ be linearly independent and let $v\notin S$. Then $S\cup\{v\}$ is linearly dependent if and only if $v\in\operatorname{span}(S)$.

I'm trying to prove the claim from the theorem. What I struggle with is that I only seem to be able to prove the claim when $T=S\cup\{v\}$, where $S$ is linearly independent and $v\notin S$. Then the theorem tells us that $v\in\operatorname{span}(S)$. This in turn implies $\operatorname{span}(S)=\operatorname{span}(S\cup\{v\})$ (see below for a proof of why this is implied by $v\in\operatorname{span}(S)$). Hence we can take the proper subset $S\subset S\cup\{v\}=T$ as the set in the claim preceding the theorem. But what if $T$ is not of the form $S\cup\{v\}$? I feel like I've only proved a very special case.

Regarding $v\in\operatorname{span}(S)\implies\operatorname{span}(S)=\operatorname{span}(S\cup\{v\})$, the inclusion $\operatorname{span}(S)\subset\operatorname{span}(S\cup\{v\})$ always holds. The reverse inclusion follows since any $w\in \operatorname{span}(S\cup\{v\})$ can be written as $w=a_1u_1+\cdots +a_nu_n+bv$, where $u_1,\ldots,u_n\in S$. Since $v\in\operatorname{span}(S)$, $w$ is actually a linear combination of vectors in $S$.

 

[PeroK]

What definition of linear independence and linear dependence are you using?

 

[psie]

What definition of linear independence and linear dependence are you using?

A subset $S$ of a vector space $\mathsf V$ is linearly dependent if there exists a finite number of distinct vectors $u_1,\ldots,u_n$ in $S$ and scalars, $a_1,\ldots,a_n$, not all zero such that $$a_1u_1+\cdots+a_nu_n=0.$$ A subset $S$ of $\mathsf V$ is linearly independent if it is not linearly dependent.

It was a bit of a mess in post #1, but I think I've figured it out.

 

[fresh_42]

The idea is to approach the concept of a basis as a maximal set of linearly independent vectors without requiring that it is finite. Add one non-zero vector to such a set and it isn't linearly independent anymore, which in return automatically generates a linear expression in terms of basis vectors. Note that such a linear expression is always finite. There are no infinite series as we in general do not have a concept of convergence! That would require a topological vector space, e.g. a metric.