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 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[change | change source]

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 .