Linear independence

In linear algebra, a set of vectors is said to be linearly independent if there exists no vector in the set that is equal to a linear combination of the other vectors in the set. If such a vector exists, then the vectors are said to be linearly dependent. Linear independence is part of the definition of linear basis.

A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space.

Definition

A sequence of vectors

v

1

,

v

2

,

…

,

v

k

{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\dots ,\mathbf {v} _{k}}

from a vector space V is said to be linearly dependent, if there exist scalars

a

1

,

a

2

,

…

,

a

k

,

{\displaystyle a_{1},a_{2},\dots ,a_{k},}

not all zero, such that

a

1

v

1

+

a

2

v

2

+

⋯

+

a

k

v

k

=

0

,

{\displaystyle a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+\cdots +a_{k}\mathbf {v} _{k}=\mathbf {0} ,}

where

0

{\displaystyle \mathbf {0} }

denotes the zero vector.

If ⁠

k

=

1

{\displaystyle k=1}

⁠, this implies that a single vector is linear dependent if and only if it is the zero vector.

If ⁠

k

>

1

{\displaystyle k>1}

⁠, this implies that at least one of the scalars is nonzero, say

a

1

≠

0

{\displaystyle a_{1}\neq 0}

, and the above equation is able to be written as

v

1

=

−

a

2

a

1

v

2

+

⋯

+

−

a

k

a

1

v

k

.

{\displaystyle \mathbf {v} _{1}={\frac {-a_{2}}{a_{1}}}\mathbf {v} _{2}+\cdots +{\frac {-a_{k}}{a_{1}}}\mathbf {v} _{k}.}

Thus, a set of vectors is linearly dependent if and only if one of them is zero or a linear combination of the others.

A sequence of vectors

v

1

,

v

2

,

…

,

v

n

{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\dots ,\mathbf {v} _{n}}

is said to be linearly independent if it is not linearly dependent, that is, if the equation

a

1

v

1

+

a

2

v

2

+

⋯

+

a

n

v

n

=

0

,

{\displaystyle a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+\cdots +a_{n}\mathbf {v} _{n}=\mathbf {0} ,}

can only be satisfied by

a

i

=

0

{\displaystyle a_{i}=0}

for

i

=

1

,

…

,

n

.