Basis (linear algebra)

Subset of a vector space that allows defining coordinates
This picture illustrates the standard basis in R2. The red and blue vectors are the elements of the basis; the green vector can be given with the basis vectors.

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 dimension of a given vector space is the number of elements of the basis.


If   is the vector space then:

  is a basis of  .

It's easy to see that for any element of   it can be represented as a combination of the above basis. Let   be any element of   and let  .

Since   and   are elements of   then they can be written as   and so on.

Then the combination equals the element  .

This shows that the set   is a basis of  .