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Vector Space Infinite Vector Finite Set Set Difference Ring Scalar Module Field Characterization Linear Combination Eigenvector

1	Scalar
1	Vector Space
1	Vector
1	Field
1	Finite
1	Set
1	Set Difference
1	Infinite
1	Ring
1	Module
2	Characterization
3	Linear Combination
5	Eigenvector

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Linear Independence

Let be a vector space over a field . We say that $v_1,\ldots, v_k\in V$ are linearly dependent if there exist scalars $\lambda_1,\ldots, \lambda_k\in F$ , not all zero, such that

$\displaystyle \lambda_1 v_1+ \cdots +\lambda_k v_k = 0 .$

If no such scalars exist, then we say that the vectors are linearly independent. More generally, we say that a (possibly infinite) subset $S\subset V$ is linearly independent if all finite subsets of

are linearly independent.

In the case of two vectors, linear dependence means that one of the vectors is a scalar multiple of the other. As an alternate characterization of dependence, we also have the following.

Proposition 1 Let $S\subset V$ be a subset of a vector space. Then, is linearly dependent if and only if there exists a $v\in S$ such that can be expressed as a linear combination of the vectors in the set $S\backslash \{v\}$ (all the vectors in other than ).

Remark. Linear independence can be defined more generally for modules over rings: if is a (left) module over a ring . A subset of is linearly independent if whenever $r_1m_1+\cdots +r_nm_n=0$ for $r_i\in R$ and $m_i\in M$ , then $r_1=\cdots =r_n=0$ .