Basis (linear Algebra) - Definition

Definition

A basis B of a vector space V over a field F is a linearly independent subset of V that spans V.

In more detail, suppose that B = { v1, …, vn } is a finite subset of a vector space V over a field F (such as the real or complex numbers R or C). Then B is a basis if it satisfies the following conditions:

  • the linear independence property,
for all a1, …, anF, if a1v1 + … + anvn = 0, then necessarily a1 = … = an = 0; and
  • the spanning property,
for every x in V it is possible to choose a1, …, anF such that x = a1v1 + … + anvn.

The numbers ai are called the coordinates of the vector x with respect to the basis B, and by the first property they are uniquely determined.

A vector space that has a finite basis is called finite-dimensional. To deal with infinite-dimensional spaces, we must generalize the above definition to include infinite basis sets. We therefore say that a set (finite or infinite) BV is a basis, if

  • every finite subset B0B obeys the independence property shown above; and
  • for every x in V it is possible to choose a1, …, anF and v1, …, vnB such that x = a1v1 + … + anvn.

The sums in the above definition are all finite because without additional structure the axioms of a vector space do not permit us to meaningfully speak about an infinite sum of vectors. Settings that permit infinite linear combinations allow alternative definitions of the basis concept: see Related notions below.

It is often convenient to list the basis vectors in a specific order, for example, when considering the transformation matrix of a linear map with respect to a basis. We then speak of an ordered basis, which we define to be a sequence (rather than a set) of linearly independent vectors that span V: see Ordered bases and coordinates below.

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