Linear and multilinear algebra; matrix theory Definition

Linear independence

Definition

Let $F$ be a field and let $V$ be a vector space over $F$ . A family $(v_1, \dots, v_k)$ of vectors of $V$ is linearly independent when the only linear combination of them that equals $0$ is the trivial one. This means that $a_1 v_1 + \cdots + a_k v_k = 0 \implies a_1 = \cdots = a_k = 0$ for all $a_1, \dots, a_k \in F$ . An arbitrary family $(v_i)_{i \in I}$ is linearly independent when each finite subfamily is linearly independent. A family that is not linearly independent is linearly dependent.

Remarks

The vectors $(1, 0)$ and $(1, 1)$ in $\R^2$ are linearly independent; the vectors $(1, 2)$ and $(2, 4)$ are linearly dependent. A family that contains the zero vector, or the same vector twice, is linearly dependent. A family of at least two vectors is linearly dependent if and only if one of its vectors is a linear combination of the others. In $F^n$ , a family of more than $n$ vectors is linearly dependent.

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Vector space definition
Linear combination definition
Linear independence definition

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Tag	`0EVHJFTG9Z`
URI	`m2://0EVHJFTG9Z`
Kind	definition
Writing	W1 stated
MSC	15A03
Defines	Linear independence
Uses	Linear combination

Linear independence

Definition

Remarks

Reading path

Where this leads

Further reading

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