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[parent] properties of linear independence (Result)

Let $ V$ be a vector space over a field $ k$. Below are some basic properties of linear independence.

  1. $ S\subseteq V$ is never linearly independent if $ 0\in S$.
    Proof. Since $ 1\cdot 0=0$. $ \qedsymbol$
  2. If $ S$ is linearly independent, so is any subset of $ S$. As a result, if $ S$ and $ T$ are linearly independent, so is $ S\cap T$. In addition, $ \varnothing$ is linearly independent, its spanning set being the singleton consisting of the zero vector 0.
    Proof. If $ r_1v_1+\cdots r_nv_n=0$, where $ v_i\in T$, then $ v_i\in S$, so $ r_i=0$ for all $ i=1,\ldots, n$. $ \qedsymbol$
  3. If $ S_1\subseteq S_2\subseteq \cdots$ is a chain of linearly independent subsets of $ V$, so is their union.
    Proof. Let $ S$ be the union. If $ r_1v_1+\cdots r_nv_n=0$, then $ v_i\in S_{a(i)}$, for each $ i$. Pick the largest $ S_{a(i)}$ so that all $ v_i$'s are in it. Since this set is linearly independent, $ r_i=0$ for all $ i$. $ \qedsymbol$
  4. $ S$ is a basis for $ V$ iff $ S$ is a maximal linear independent subset of $ V$. Here, maximal means that any proper superset of $ S$ is linearly dependent.
    Proof. If $ S$ is a basis for $ V$, then it is linearly independent and spans $ V$. If we take any vector $ v\notin S$, then $ v$ can be expressed as a linear combination of elements in $ S$, so that $ S\cup \lbrace v\rbrace$ is no longer linearly independent, for the coefficient in front of $ v$ is non-zero. Therefore, $ S$ is maximal.

    Conversely, suppose $ S$ is a maximal linearly independent set in $ V$. Let $ W$ be the span of $ S$. If $ W\ne V$, pick an element $ v\in V-W$. Suppose $ 0=r_1v_1+\cdots r_nv_n+rv$, where $ v_i\in S$, then $ -rv = r_1v_1+\cdots +r_nv_n$. If $ r\ne 0$, then $ v$ would be in the span of $ S$, contradicting the assumption. So $ r=0$, and as a result, $ r_i=0$, since $ S$ is linearly independent. This shows that $ S\cup \lbrace v\rbrace$ is linearly independent, which is impossible since $ S$ is assumed to be maximal. Therefore, $ W=V$. $ \qedsymbol$

Remark. All of the properties above can be generalized to modules over rings, except the last one, where the implication is only one-sided: basis implying maximal linear independence.



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Cross-references: implication, rings, modules, properties, conversely, coefficient, linear combination, vector, spans, linearly dependent, proper superset, independent, iff, basis, union, chain, zero vector, singleton, spanning set, addition, subset, linearly independent, field, vector space
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This is version 2 of properties of linear independence, born on 2008-05-28, modified 2008-05-28.
Object id is 10633, canonical name is PropertiesOfLinearIndependence.
Accessed 413 times total.

Classification:
AMS MSC: 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)
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