Graphics Reference
In-Depth Information
There's another equally useful interpretation of
Av
.Ifwelet
b
i
denote the
columns
of
A,
then
Av
=
v
1
b
1
+
v
2
b
2
+
...
+
v
n
b
n
,
(7.58)
that is, the product of
A
with
v
is a linear combination of the columns of
A.
One particularly useful consequence of the definition of
Av
is that if we want
to multiply
A
by several vectors
v
1
,
v
2
, ...,
v
k
, we can express this product by
concatenating the vectors
v
i
into a matrix
V
whose columns are
v
1
,
v
2
, etc.; the
product
AV
(7.59)
is then a matrix
W
whose
i
th column is
Av
i
. This is no more computationally
efficient than multiplying
A
by each individual vector. The key application of this
idea is when we have one set of vectors
v
1
,
v
2
,
...
,
v
k
and a second set of vectors
w
1
,
w
2
,
...
,
w
k
, and we wish to find a matrix with
Av
i
=
w
i
for
i
=
1,
...
,
k
, that
is, we wish to have
AV
=
W.
(7.60)
Often it's impossible to achieve exact equality, but it will turn out that we can find
the “best” such matrix
A
by straightforward operations on the matrices
V
and
W,
but we'll wait to present the main ideas in context in Section 10.3.9.
In general, matrix multiplication is not commutative:
AB
=
BA.
Inline Exercise 7.3:
(a) If
A
is a 2
×
3 matrix and
B
is a 3
×
1 matrix, show
that
AB
makes sense, but
BA
does not.
(b) Let
A
=
123
T
and
B
=
011
. Compute both
AB
and
BA.
The spaces
R
2
and
R
3
that we've been discussing, and
R
n
in general, have
certain properties. There's a notion of addition (which is commutative and asso-
ciative), and of scalar multiplication (again associative, and distributive over addi-
tion). There's also a zero vector with the property that
0
+
v
=
v
+
0
=
v
for
any vector
v
. And there are additive inverses: Given a vector
v
, we can always
find another vector
w
with
w
+
v
=
0
. These properties, taken together, make
R
n
a vector space. There are a few other vector spaces we'll encounter in graphics;
some of these will arise in our discussion of images, others in our discussion of
splines, and still others in our discussion of rendering. Most have a common form:
They are spaces whose elements are all
functions
.
Recall that, if we had a vector
w
R
2
, we built a function
∈
φ
w
:
R
2
→
R
:
v
→
w
·
v
.
(7.61)
This is a linear function on
R
2
, and in fact,
any
linear function from
R
2
to
R
has
this particular form.
