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

Let be a field, and let be a -vector space. Let denote a family of vectors in . Then the vector

is called a *linear combination* of this vectors

*coefficient tuple*).

Two different coefficient tuples can define the same vector.

Let be a
field,
and let be a
-vector space.
A family
, ,
is called a *generating system*
(or *spanning system*)
of , if every vector
can be written as

with a finite subfamily , and with

.In , the standard vectors , , form a generating system. In the polynomial ring , the powers , , form an (infinite) generating system.

Let be a field, and let be a -vector space. For a family , , we set

*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 . A single vector spans the space
.
For
,
this is a *line*, a term we will make more precise in the framework of dimension theory. For two vectors
and ,
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
,
then is superfluous, and the linear subspace generated by the two vectors equals the linear subspace generated by . If this is not the case
(and
and
are not ),
then the two vectors span a "plane“.

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

Let be a field, and let be a

-vector space. Then the following statements hold.- For a family , , of elements in , the linear span is a linear subspace of .
- The family
, ,
is a spanning system of , if and only if

### Proof

- ↑ This follows from the definition, if we use the convention that the empty sum equals .