Bases

A basis of a vector space is a finite set of vectors in that space which

  • span the whole space
  • are linearly independent.

Examples. Several of the spaces you've learned about have what are called standard bases:

The standard basis for two-dimensional Euclidean space is {i, j}.

The standard basis for three-dimensional Euclidean space is {i, j, k}.

The standard basis for Rn is the set of elementary vectors {e1, e2, ..., en}.

The standard basis for the space Pn is the set of polynomials {1, x, x2, ..., xn}.

The standard basis for the space Mnm is the set of m x n matrices with one entry 1 and all the others 0.

Each of these spaces has many more bases - in Euclidean three-space, for example, any set of three vectors not all in a plane forms a basis.

 

A set of vectors, linearly independent or not, which spans a space can represent any vector in the space, possibly in several different ways. Once we have a linearly independent spanning set, however, there is only one way to represent each vector.

If {b1, b2, ..., bn} is a basis of a vector space V, then any vector in V can be expressed as a linear combination of b1, b2, ..., bn in only one way.

Proof. Suppose you had what you thought were two ways of representing the vector v as a linear combination of b1, b2, ..., bn:

v = c1b1 + c2b2 + ... + cnbn

v = d1b1 + d2b2 + ... + dnbn.

Subtract to get

0 = (c1 – d1)b1 + (c2 – d2)b2 + ... + (cn – dn)bn.

But this is a linearly dependence relation for the basis vectors. Since basis vectors are linearly independent, all the coefficients of this relation must be 0, so c1 = d1, c2 = d2, ..., cn = dn. So in fact, the two ways of representing the vector v turned out to be the same.

This means we can talk about the way of representing a vector in terms of a basis instead of a way.

For any vector v in a space with a basis B = {b1, b2, ..., bn}, the unique coefficients c1, c2, ..., cn with v = c1b1 + c2b2 + ... + cnbn are called the coordinates (or components) of the vector v with respect to the basis B. The n-tuple (v)B = (c1, c2, ..., cn) is called the coordinate vector of v with respect to B.