# Linear Span

The term **linear span** refers to a set of n vectors v_{1}, v_{2}, ..., v_{n} in a vector space V over a field K, which generates a subspace of V through linear combinations with scalars α_{1}, α_{2}, ..., α_{n} from K. $$ Span( v_1 ,..., v_n ) = \{ α_1 \cdot v_1 ,..., α_n \cdot v_n \: \: \forall α_i \in K \} $$

A linear span is also denoted by the symbols L_{k}, L or <>.

$$ L_k ( v_1 ,..., v_n ) $$ $$ L( v_1 ,..., v_n ) $$ $$ < v_1 ,..., v_n > $$

The Span(v_{1}, ..., v_{n}) forms a subspace of the vector space V.

It is known as the **subspace generated by vectors v _{1}, ..., v_{n}**.

## Practical Examples

__Example 1__

If a vector v_{1} in the vector space V over the field R^{2} is non-zero ( v_{1}≠Ø ), then its linear span L(v_{1}) comprises all vectors α_{1}v_{1 }for every α_{1}∈R, lying on the line through the origin in the direction of v_{1}.

Each value of α_{1}∈R generates a vector α_{1}v_{1 }on the line L_{k}, in the same direction but of varying length, each a multiple of v_{1}.

It's immediately apparent that L_{k} is a vector subspace of V.

__Example 2__

Given two vectors v_{1} and v_{2} in the vector space V over the field R^{2}, the linear span L_{k}(v_{1}, v_{2}) consists of all vectors α_{1}v_{1 }+ α_{2}v_{2 }for every α_{1}, α_{2}∈R originating in the plane.

Each pair of values α_{1}, α_{2}∈R generates a vector w= α_{1}v_{1} + α_{2}v_{2} with different direction, orientation, and length.

If the two vectors are non-parallel, the vector w is determined by the parallelogram rule as a sum of a multiple of the vector v_{1} and a multiple of the vector v_{2}.

Similarly, in this scenario, L_{k}(v_{1}+v_{2}) forms a vector subspace of V.

**Note**. If the two vectors are parallel, it reverts to the case of the line through the origin. Here, the two vectors share the same direction and lie on the same line . Thus α_{1}v_{1}+α_{2}v_{2 }are multiples of the same vector. It’s akin to writing α_{1}v_{1}. In fact, if the vectors are parallel, we have L(v_{1},v_{2}) = L(v_{1}).