[선형대수학] 선형 독립인 부분집합이 있을 때, 항상 이를 포함하는 극대 선형 독립 부분집합이 있다. (Existence of Maximal Linear Independent Subset) 모든 벡터공간은 기저를 갖는다.

$\underline{Thm}$ (By the axiom of choice)

Let $S$ be  a linear independent subset of a vector space $V$.

Then there exists a maximal linear independent subset $\beta$ of $V$ with $S \subset \beta$.

 

$\underline{Proof}$

Let $\mathcal{F}$ be the set of all linear independent subsets of $V$ containing $S$.

Let $\mathcal{C} \subset \mathcal{F}$ be any chain.

Define $\displaystyle L_{\mathcal{C}}$ = $\underset{A \in \mathcal{C}}{\cup A}$. Then $A \subset L_{\mathcal{C}} \; \forall A \in \mathcal{C}$.

We check $L_{\mathcal{C}} \in \mathcal{F}$. Since $S \subset A\;\forall A \in \mathcal{C}$, we have $S \subset L_{\mathcal{C}}$.

It remains to check that $L_{\mathcal{C}}$ is linear independent.

Suppose $u_{1},\cdots,u_{n} \in L_{\mathcal{C}},\; a_{1},\cdots,a_{n} \in \mathcal{F}$. and $a_{1}u_{1}+\cdots+a_{n}u_{n}=0$.

Then $\forall i \in \{1,\cdots,n \}$, $u_{i} \in A_{i}$ for some $A_{i} \in \mathcal{C}$.

Since $\mathcal{C}$ is a chain in $\mathcal{F}$, $\exists k \in \{ 1, \cdots, n \}$ such that $u_{i} \in A_{i} \subset A_{k} \; \forall i \in \{ 1, \cdots, n \}$.

Since $A_{k}$ is linear independent, we have $a_{1}=\cdots=a_{n}=0$. 

Thus, $L_{\mathcal{C}}$ is linear independent, so that $L_{\mathcal{C}}=\mathcal{F}$.

By Zorn's lemma, $\mathcal{F}$ has a maximal set, say $\beta$. This $\beta$ is a desired set.

 

$\underline{Coro}$

Every vector space has a basis.