Basis (linear algebra)

In linear algebra, a basis is a set of vectors in a given vector space with certain properties:

 One can get any vector in the vector space by multiplying each of the basis vectors by different numbers, and then adding them up.
 If any vector is removed from the basis, the property above is no longer satisfied. 

The plural of basis is bases. For any vector space , any basis of  will have the same number of vectors. This number is called the dimension of .

Example
 is a basis of  as a vector space over .

Any element of  can be written as a linear combination of the above basis. Let  be any element of  and let . Since  and  are elements of , then we can write . So  can be written as a linear combination of the elements in .

Also, this process would not be possible for any vector  if an element was removed from . So  is a basis for .

The basis  is not unique; there are infinitely many bases for . Another example of a basis would be .

Category:Linear algebra