Basis (linear algebra)

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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.

Example[change | change source]

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 .