Linear Algebra

206 Chapter 3. Maps Between Spaces
then h1 : R 2 → R 2 represented by H with respect to B1,D1 maps
� � � �
c1 c1
=
c2
c2
B1
↦→
� �
c1
0
D1
=
� �
c1
0
while h2 : R 2 → R 2 represented by H with respect to B2,D2 is this map.
� � � �
c1 c2
=
c2
c1
B2
↦→
� �
c2
0
D2
� �
0
=
These two are different. The first is projection onto the x axis, while the second
is projection onto the y axis.
So not only is any linear map described by a matrix but any matrix describes
a linear map. This means that we can, when convenient, handle linear maps
entirely as matrices, simply doing the computations, without have to worry that
a matrix of interest does not represent a linear map on some pair of spaces of
interest. (In practice, when we are working with a matrix but no spaces or
bases have been specified, we will often take the domain and codomain to be R n
and R m and use the standard bases. In this case, because the representation
is transparent—the representation with respect to the standard basis of �v is
�v—the column space of the matrix equals the range of the map. Consequently,
the column space of H is often denoted by R(H).)
With the theorem, we have characterized linear maps as those maps that act
in this matrix way. Each linear map is described by a matrix and each matrix
describes a linear map. We finish this section by illustrating how a matrix can
be used to tell things about its maps.
2.3 Theorem The rank of a matrix equals the rank of any map that it represents.
Proof. Suppose that the matrix H is m×n. Fix domain and codomain spaces
V and W of dimension n and m, with bases B = 〈 � β1,..., � βn〉 and D. Then H
represents some linear map h between those spaces with respect to these bases
whose rangespace
{h(�v) � � �v ∈ V } = {h(c1 � β1 + ···+ cn � βn) � � c1,...,cn ∈ R}
= {c1h( � β1)+···+ cnh( � βn) � � c1,...,cn ∈ R}
is the span [{h( � β1),...,h( � βn)}]. The rank of h is the dimension of this rangespace.
The rank of the matrix is its column rank (or its row rank; the two are
equal). This is the dimension of the column space of the matrix, which is the
span of the set of column vectors [{Rep D(h( � β1)),...,Rep D(h( � βn))}].
To see that the two spans have the same dimension, recall that a representation
with respect to a basis gives an isomorphism Rep D : W → R m . Under
this isomorphism, there is a linear relationship among members of the rangespace
if and only if the same relationship holds in the column space, e.g, �0 =
c2