Basis and Dimension
In this section we learn how to equip an arbitrary subvector space 
of 
with a coordinate system. To specify the coordinate axes, we use a collection of vectors 
in which spans and is linearly independent. Such a collection of vectors is called a basis of . We will discuss the use of bases in detail in the next section. Here we merely introduce the concept and we learn how to find a basis for .
- spans ;
- is linearly independent.
Every vector space containing a nonzero vector has a basis; in fact it has infinitely many. However, any two bases of have the same number of vectors. This number of vectors in a basis of is called the dimension of .
of isdim() 
, if has a basis of vectors.
{0}, we set dim() 0. 
The dimension of a subvector space of is a measure of the size of . The properties of "dimension" agree nicely with our intuition of "size". For example:
- There are many low dimensional subspaces inside a high dimensional one.
- However, a high dimensional space cannot sit inside a low dimensional one; and ...
- if sits inside
, and dim() dim(), then . - If and are subspaces of whose intersection contains only the 0-vector and which satisfy span(
) , then dim() + dim() 
.
This completes our exposition on the general theory of bases.
Let us now turn to the specific task of finding a basis for the subvector space which results from any one of the subvector space constructions row space, column space, span, null space, and orthogonal complement we have introduced earlier.
be a matrix of size (
,), and let 
denote its reduced row echelon form.- A basis for the row space of is given by the nonzero rows of ; therefore dim() rank().
- A basis for the column space of is given by selecting from the column vectors from those positions in which has a leading 1. Therefore, dim() rank().
- A basis for the null space of can be calculated from ; therefore dim null()
rank().
As a Summary Given vectors a
, ... ,a
- use them as row vectors of a matrix and apply 1) above to find a basis for span{a, ... ,a};
- use them as the column vectors of matrix and apply 2) above to select a basis for span{a, ... ,a} from amongst the given vectors.
- use them as the row vectors of a matrix and apply 3) above to find a basis for the orthogonal complement {a, ... ,a}.
Exercises on basis and dimension
We close this section with some results for future reference, including two alternate characterizations of a basis.
If
span(
), then is a basis of if and only if removing any vector v from yields a set ' which fails to span .
A linearly independent subset
of is a basis of if and only if for every vector v in , 
{v} is not linearly independent.
, ... ,a
be linearly independent vectors in , and let x (
, ... ,), ... ,x (, ... ,) be a basis of 
. Then
| y |
|
a + ... + a |
| ¦ | ¦ ¦ | |
| y |
|
a + ... + a |
, ... ,a}.