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.

Read more about this topic:  Basis (linear Algebra)

Famous quotes containing the word definition:

    Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.
    The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)

    Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.
    Elaine Heffner (20th century)

    ... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.
    Adrienne Rich (b. 1929)