A four dimensional vector space V has a basis  eeee 1 , 2 , 3 , 4  . A vector vV  has the

2.6.6 A four dimensional vector space V has a basis  eeee 1 , 2 , 3 , 4  . A vector vV  has the

component representation

v  5 e 1  3 e 2  6 e 3  2 e 4 (2.6.50)

You are also given a change of basis to a new basis for V , defined by

e 1  e ˆ 2  e ˆ 3 3 e ˆ 4

e 2  e ˆ 1 e ˆ 2 e ˆ 3 2 e ˆ 4

Determine the transition matrix associated with this basis change. Also, determine the components

of the vector v with respect to the basis  eeee ˆˆˆˆ 1 , 2 , 3 , 4  of V .

Sec. 2.7 • Image Space, Rank and Kernel of a Matrix

Section 2.7. Image Space, Rank and Kernel of a Matrix

The ideas introduced in Sections 2.1 through 2.7 contain useful information relative to the problem of solving the matrix equation (1.2.1), repeated,

Equivalently, we can write (2.7.1) in its matrix form equation (1.2.2), repeated,

 A 11 A 12  A 1 N  x 1 b 1

or, equivalently, as (1.2.3), repeated,

A x  b (2.7.3)

where, as usual, A is the matrix

The matrix A is an element of the vector space M MN  , the column matrix x is a member

M N  1 M  of the vector space 1 and b is a member of the vector space M . Recall from Section 2.5 M MN 

N  1 M  that dim 1  MN , dim M  N and dim M  M . In equation (1.8.3), we stressed the MN view of  A M as a function

174 Chap. 2 •

VECTOR SPACES

N  1 M  1 A : M  M (2.7.5)

M N  As a function whose domain is the vector space 1 and whose values lie in the vector space M M  1 , the usual matrix operations imply

A ( v 1  v 2 )  A v 1  A v 2 (2.7.6)

M N  for all vectors 1 vv 1 , 2  and and

A (  v )   A () v (2.7.7)

N  for all vectors 1 v  M and  C . Functions defined on vector spaces that obey rules like (2.7.6)

MN and (2.7.7) are called linear transformations. The matrix  A M is but one example of a linear

transformation. These functions will be studies in greater generality in Chapter 3. In this section, we are interested in recording properties of this particular kind of linear transformation.

As explained in Section 1.8, the range of the function A is the set of all values of the function. In other words, the range is the set of possible values of Ax generated for all possible M N  values of x in 1 . In Section 1.8, we gave this quantity the symbol RA  . It was defined

formally in equation (1.8.4), repeated,

N  1 RA   

A xx  M  (2.7.8)

The following figure should be helpful.

RA 

Sec. 2.7 • Image Space, Rank and Kernel of a Matrix

M M  It is a fact that the set 1 RA  is a subspace of . The proof of this assertion, like all such

assertions about subspaces, simply requires that the definition of subspace be satisfied. If v 1 and

2 are two members of M , then Av 1 and Av 2 are members of RA  . We need to prove that

their sum, A v 1  A v 2 , is also in RA  . The proof follows from (2.7.6), repeated,

A v 1  A v 2  A  v 1  v 2  (2.7.9)

N  Since 1 v

1  v 2  M , A  v 1  v 2   RA  and, by (2.7.9), A v 1  A v 2  RA  . Thus, the first part

of the definition of a subspace is established. An entirely similar manipulation establishes that

  A v  RA  for all  , and, thus, RA  is a subspace. In order to stress the fact that the range is

a subspace, we shall begin to refer to the range as the image space. There are two other important

M N  1 M  concepts involving the matrix 1 A :  M that we will now introduce.

M  Definition 1 : A matrix A :  is said to be onto if it has the property that RA  M

M N  1 M M  Definition 1 : A matrix A :  is one to one if

A v 1  A v 2 implies v 1  v 2 (2.7.10)

Onto and one to one matrices have special properties which we will characterize later in this section.

In Section 1.8, we assigned the columns of the matrix (2.7.4) the symbols a j , j  1, 2,..., N ,

by the formulas

for

j  1,..., N (2.7.11)

  A Mj  

M  The set of column vectors 1 

aa 1 , 2 ,..., a N  consists of vectors in the set M . As established in

Section 1.8, for an arbitrary vector x  M N  1 , equation (1.8.7), repeated, tells us that

A x  a 11 x  a 2 x 2  a 3 x 3    a N x N (2.7.12)

176 Chap. 2 •

VECTOR SPACES

Equation (2.7.12) establishes that every vector in the image space RA  has the representation as a

linear combination of the vectors in the set  aa 1 , 2 ,..., a N  . This fact is summarized by the formula

RA   Span  aa 1 , 2 ,..., a N  (2.7.13)

Because of the result (2.7.13), the image space RA  is also known as the column space. We next defined the rank of the matrix A .