Basis (linear algebra)

In mathematics, a set B of elements of a vector space V is called a basis (pl.: bases) if every element of V can be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B. The elements of a basis are called basis vectors.

Equivalently, a set B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B. In other words, a basis is a linearly independent spanning set.

A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space.

This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.

Basis vectors find applications in the study of crystal structures and frames of reference.

Definition

A basis B of a vector space V over a field F (such as the real numbers

R

{\displaystyle \mathbb {R} }

or the complex numbers

C

{\displaystyle \mathbb {C} }

) is a linearly independent subset of V that spans V. This means that a subset B of V is a basis if it satisfies the two following conditions:

linear independence: for every finite subset

{

v

1

,

…

,

v

m

}

{\displaystyle \{\mathbf {v} _{1},\dotsc ,\mathbf {v} _{m}\}}

of B, if

c

1

v

1

+

⋯

+

c

m

v

m

=

0

{\displaystyle c_{1}\mathbf {v} _{1}+\cdots +c_{m}\mathbf {v} _{m}=\mathbf {0} }

for some

c

1

,

…

,

c

m

{\displaystyle c_{1},\dotsc ,c_{m}}

in F, then

c

1

=

⋯

=

c

m

=

0

{\displaystyle c_{1}=\cdots =c_{m}=0}

; and

the spanning property: for every vector

v

{\displaystyle \mathbf {v} }

in

V

{\displaystyle V}

, one can choose

a

1

,

…

,

a

n

{\displaystyle a_{1},\dotsc ,a_{n}}

in

F

{\displaystyle F}

and

v

1

,

…

,

v

n

{\displaystyle \mathbf {v} _{1},\dotsc ,\mathbf {v} _{n}}

in

B

{\displaystyle B}

such that

v

=

a

1

v

1

+

⋯

+

a

n

v

n

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

. In other words,

v

{\displaystyle \mathbb {v} }

can be represented as a linear combination of some vectors in

B

{\displaystyle B}

.

The first property may be equivalently phrased as follows: if a linear combination of vectors in B is equal to the zero vector, then all the scalars coefficients of the combination must be zero.

If B is a basis for V, then every vector

v

{\displaystyle \mathbf {v} }

in V can be written as a linear combination of vectors in B (by the spanning property); it follows from linear independence that this can be done in exactly one way.