Basis (linear Algebra) - Properties

Properties

Again, B denotes a subset of a vector space V. Then, B is a basis if and only if any of the following equivalent conditions are met:

• B is a minimal generating set of V, i.e., it is a generating set and no proper subset of B is also a generating set.
• B is a maximal set of linearly independent vectors, i.e., it is a linearly independent set but no other linearly independent set contains it as a proper subset.
• Every vector in V can be expressed as a linear combination of vectors in B in a unique way. If the basis is ordered (see Ordered bases and coordinates below) then the coefficients in this linear combination provide coordinates of the vector relative to the basis.

Every vector space has a basis. The proof of this requires the axiom of choice. All bases of a vector space have the same cardinality (number of elements), called the dimension of the vector space. This result is known as the dimension theorem, and requires the ultrafilter lemma, a strictly weaker form of the axiom of choice.

Also many vector sets can be attributed a standard basis which comprises both spanning and linearly independent vectors.

Standard bases for example:

In Rn {E1,...,En} where En is the n-th column of the identity matrix which consists of all ones in the main diagonal and zeros everywhere else. This is because the columns of the identity matrix are linearly independent can always span a vector set by expressing it as a linear combination.

In P₂ where P₂ is the set of all polynomials of degree at most 2 {1,x,x2} is the standard basis.

In M₂₂ {M_1,1,M_1,2,M_2,1,M_2,2} where M₂₂ is the set of all 2×2 matrices. and M_m,n is the 2×2 matrix with a 1 in the m,n position and zeros everywhere else. This again is a standard basis since it is linearly independent and spanning.