Span And Linear Independence

Linear Combination

A linear combination of the vectors

{v_{1}, \dots, v_{m}}

of vectors in a vector space

V

is:

a_{1} v_{1} + \dots + a_{m} v_{m}

where

a_{1}, \dots, a_{m} \in 𝔽

Span

The set of all linear combinations of the vectors

{v_{1}, \dots, v_{m}} \subseteq V

(where

V

is a vector space) called the span of

{v_{1}, \dots, v_{m}}

and we define the notation:

span ({v_{1}, . . ., v_{m}}) : = {a_{1} v_{1} + \dots + a_{m} v_{m} : a_{1}, . . ., a_{m} \in 𝔽}

we define

span (\emptyset) : = {0}

Linearly Independent

A set of vectors

{v_{1}, \dots, v_{m}} \subseteq V

is said to be linearly independent if the only solution to

\sum_{i = 1}^{m} a_{i} v_{i} = 0

\forall i \in {1, . . ., m}, a_{i} = 0

. We define

\emptyset

to be linearly independent.

General Linear Independence

A set

S \subseteq V

is said to be generally linearly independent if for every every finite subset

F \subseteq S

F

is linearly indepenent

Spanning Set

Suppose that

V

is a vector space, then we say that

S

is a spanning set of $V$ if

span (S) = V

Linearly Independent iff Unique Representation

{v_{1}, \dots, v_{m}}

is linearly independent iff each vector in the set

{v_{1}, \dots, v_{m}}

has only one representation as a linear combination of

{v_{1}, \dots, v_{m}}

Linearly Dependent

{v_{1}, \dots, v_{m}} \subseteq V

is said to be linearly dependent if it is not linearly independent

Linearly Depedent iff Zero has a Non-Trivial Representation

{v_{1}, \dots, v_{m}} \subseteq V

is linearly dependent iff there exists

a_{1}, \dots, a_{m} \in 𝔽

and

i \in [1, m]

where

a_{i} \neq 0

such that

\sum_{i = 1}^{m} a_{i} v_{i} = 0