# Vector space/Linear combination/Generating system/Introduction/Section

## Definition

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ be a ${\displaystyle {}K}$-vector space. Let ${\displaystyle {}v_{1},\ldots ,v_{n}}$ denote a family of vectors in ${\displaystyle {}V}$. Then the vector

${\displaystyle s_{1}v_{1}+s_{2}v_{2}+\cdots +s_{n}v_{n}{\text{ with }}s_{i}\in K}$

is called a linear combination of this vectors

(for the coefficient tuple ${\displaystyle {}(s_{1},\ldots ,s_{n})}$).

Two different coefficient tuples can define the same vector.

## Definition

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ be a ${\displaystyle {}K}$-vector space. A family ${\displaystyle {}v_{i}\in V}$, ${\displaystyle {}i\in I}$, is called a generating system (or spanning system) of ${\displaystyle {}V}$, if every vector ${\displaystyle {}v\in V}$ can be written as

${\displaystyle {}v=\sum _{j\in J}s_{j}v_{j}\,,}$

with a finite subfamily ${\displaystyle {}J\subseteq I}$, and with

${\displaystyle {}s_{j}\in K}$.

In ${\displaystyle {}K^{n}}$, the standard vectors ${\displaystyle {}e_{i}}$, ${\displaystyle {}1\leq i\leq n}$, form a generating system. In the polynomial ring ${\displaystyle {}K[X]}$, the powers ${\displaystyle {}X^{n}}$, ${\displaystyle {}n\in \mathbb {N} }$, form an (infinite) generating system.

## Definition

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ be a ${\displaystyle {}K}$-vector space. For a family ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$, we set

${\displaystyle {}\langle v_{i},\,i\in I\rangle ={\left\{\sum _{i\in J}s_{i}v_{i}\mid s_{i}\in K,\,J\subseteq I{\text{ finite subset}}\right\}}\,,}$
and call this the linear span of the family, or the generated linear subspace.

The empty set generates the null space.[1] The null space is also generated by the element ${\displaystyle {}0}$. A single vector ${\displaystyle {}v}$ spans the space ${\displaystyle {}Kv={\left\{sv\mid s\in K\right\}}}$. For ${\displaystyle {}v\neq 0}$, this is a line, a term we will make more precise in the framework of dimension theory. For two vectors ${\displaystyle {}v}$ and ${\displaystyle {}w}$, the "form“ of the spanned space depends on how the two vectors are related to each other. If they both lie on a line, say ${\displaystyle {}w=sv}$, then ${\displaystyle {}w}$ is superfluous, and the linear subspace generated by the two vectors equals the linear subspace generated by ${\displaystyle {}v}$. If this is not the case (and ${\displaystyle {}v}$ and ${\displaystyle {}w}$ are not ${\displaystyle {}0}$), then the two vectors span a "plane“.

We list some simple properties for generating systems and linear subspaces.

## Lemma

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ be a

${\displaystyle {}K}$-vector space. Then the following statements hold.
1. For a family ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$, of elements in ${\displaystyle {}V}$, the linear span is a linear subspace of ${\displaystyle {}V}$.
2. The family ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$, is a spanning system of ${\displaystyle {}V}$, if and only if
${\displaystyle {}\langle v_{i},\,i\in I\rangle =V\,.}$

### Proof

${\displaystyle \Box }$
1. This follows from the definition, if we use the convention that the empty sum equals ${\displaystyle {}0}$.