Graphics Reference
In-Depth Information
()
=
T
pp
A
.
(C.2b)
If we had chosen the transpose of A, then
TA
T
T
()
=
(
)
.
pp
The distinction between choosing A or its transpose is therefore a question of whether
we want to pre- or post-multiply vectors by matrices. It does not matter which one
chooses as long as one is
consistent
. In order to have the matrix product agree with
the action of the map, the reader needs to take note of the following:
Our choice of matrices is such that one must always pre-multiply vectors!
In this way we avoid excessive transpose operations in the writing of formulas. (They
would be needed because there is a difference between a row and a column vector.
Our vectors are row vectors.)
Important Note!
Unless stated otherwise, the matrix for a linear transformation
T:
R
n
Æ
R
n
will always be defined
with respect to the standard basis
. Furthermore,
with this assumption one can then use equation (C.2b) to define an
unambiguous
bijective correspondence
between matrices and such linear transformations. This is
the correspondence we will have in mind if we have the need to pass back and forth
between matrices and transformations. A similar comment applies to transformations
T:k
n
Æ k
n
for some field k.
C.3.5. Theorem.
Let T, T
1
, T
2
:
V
Æ
V
be linear transformations and assume that A,
A
1
, and A
2
are the matrices for T, T
1
, and T
2
, respectively, with respect to some fixed
basis for
V
.
(1) The matrix for T
-1
is A
-1
.
(2) if S = T
1
T
2,
then A
2
A
1
is the matrix for S.
Proof.
The proofs follow by straightforward computations. Note though that
because of our conventions the matrices in (2) are listed in the opposite order from
that of the transformations!
Definition.
If A is the matrix for the linear transformation T, then the determinant
of A is called the
determinant
of T and is denoted by det(T). The trace of A is called
the
trace
of T and is denoted by tr(T). The rank of A is called the
rank
of T.
C.3.6. Theorem.
The determinant, trace, and rank of a linear transformation
T:
V
Æ
V
depends only on T and not the choice of basis for
V
.
Proof.
The theorem is an easy consequence of Theorem C.3.4, property (6) of deter-
minants in Theorem C.3.1, and property (2) of the trace function in Theorem C.3.2.
C.3.7. Theorem.
A linear transformation T :
V
Æ
V
is nonsingular if and only if
det(T) π 0.
Proof.
See [John67].
