Page 199 - Matrix Analysis & Applied Linear Algebra

194 Chapter 4 Vector Spaces
4.4 BASIS AND DIMENSION

Recall from §4.1 that S is a spanning set for a space V if and only if every
vector in V is a linear combination of vectors in S. However, spanning sets
can contain redundant vectors. For example, a subspace L defined by a line
2
through the origin in may be spanned by any number of nonzero vectors
{v 1 , v 2 ,..., v k } in L, but any one of the vectors {v i } by itself will suffice.
3
Similarly, a plane P through the origin in can be spanned in many different
ways, but the parallelogram law indicates that a minimal spanning set need only
be an independent set of two vectors from P. These considerations motivate the
following definition.

Basis

A linearly independent spanning set for a vector space V is called a
basis for V.

It can be proven that every vector space V possesses a basis—details for
m
the case when V⊆ are asked for in the exercises. Just as in the case of
spanning sets, a space can possess many different bases.
Example 4.4.1

n n
• The unit vectors S = {e 1 , e 2 ,..., e n } in are a basis for . This is
n
called the standard basis for .
• If A is an n × n nonsingular matrix, then the set of rows in A as well as
n
the set of columns from A constitute a basis for . For example, (4.3.3)
insures that the columns of A are linearly independent, and we know they
n n
span because R (A)= —recall Exercise 4.2.5(b).
• For the trivial vector space Z = {0}, there is no nonempty linearly indepen-
dent spanning set. Consequently, the empty set is considered to be a basis
for Z.

2 n
• The set 1,x,x ,...,x is a basis for the vector space of polynomials
having degree n or less.
2
• The infinite set 1,x,x ,... is a basis for the vector space of all polynomi-
als. It should be clear that no finite basis is possible.