Basis (linear algebra)
subset of a vector space that allows defining coordinates
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 .
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 .