[선형대수학] 선형 종속 & 독립 (Linear dependence & independence)

$\underline{Def}$ $(Linear\;Dependence \; \& \;Independence)$

A subset $S$ of a vector space $V$ is linear dependent if there are finitely many distince $u_{1}, \cdots, u_{n} \in S$ and scalars $a_{1}, \cdots, a_{n}$, not all zero, such that $a_{1}u_{1}+ \cdots + a_{n}u_{n}=0$.

A subset $S$ of a vector space $V$ is linear independent if it is not linear dependent.

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$\underline{Facts}$

$(1)$ $\phi$ is linear independent.

$(2)$ Any subset of $S$ of $V$ containing $0$ is linear dependent, since $1 \times 0 = 0$.

$(3)$ Any subset $S$ of $V$ consisting of a single non-zero vector is linear independent.

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$\underline{Prop}$

Let $S$ be a subset of a vector space $V$. Then $S$ is linear independent iff $n \in \mathbb{N}, u_{1}, \cdots, u_{n} \in S, a_{1}, \cdots, a_{n} \in \mathbb{F}$ and $a_{1}u_{1}+ \cdots +a_{n}u_{n}=0$.

$\Rightarrow a_{1}= \cdots =a_{n}=0$.

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$\underline{Def}$

We say that finitely many distinct vectors $u_{1}, \cdots, u_{n}$ in a vector space $V$ are linear dependent if the set $u_{1}, \cdots, u_{n}$ is linear dependent.

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$\underline{Note}$

A finite subset ${u_{1}, \cdots, u_{n}}$ of a vector space $V$ is linear dependent iff $\exists a_{1}, \cdots, a_{n} \in \mathbb{F}$, not all zero, such that $a_{1}u_{1}+ \cdots a_{n}u_{n}=0$.