This picture illustrates the standard basis
. 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 .