Graphics Reference
In-Depth Information
Some basic results on the diagonalization of maps and matrices in Section 1.8 lead
to a discussion of bilinear maps and quadratic forms in Section 1.9. Section 1.10
describes a general version of the three-dimensional cross product. Finally, Section
1.11 defines the generalized inverse of a transformation and matrix along with several
applications.
1.2
Lines
Our first goal in this chapter is to characterize linear subspaces of Euclidean space
and summarize some basic facts about them. There is not much to say about points,
the 0-dimensional linear subspaces, but the one-dimensional subspaces, namely,
“straight” lines, are a special case that is worth looking at separately.
First of all, let us consider lines in the plane. The usual definition of a line in this
case is as the set of solutions to a linear equation.
(The
equation
definition of a line in the plane) Any set
L
in
R
2
of the form
Definition.
{
(
)
+=
(
)
π
(
)
}
x y
,
ax
by
c
,
a b
,
00
,
,
(1.1)
where a, b, and c are fixed real constants, is called a
line
. If a = 0, then the line is
called a
horizontal line
. If b = 0, then the line is called a
vertical line
. If b π 0, then
-a/b is called the
slope
of the line.
Although an equation defines a unique line, the equation itself is not uniquely
defined by a line. One can multiply the equation for a line by any nonzero constant
and the resultant equation will still define the same line. See Exercise 1.2.1.
The particular form of the equation in our definition for a line is a good one from
a theoretical point of view, but for the sake of completeness we list several other well-
known forms that are often more convenient.
The
slope-intercept form:
The line with slope m and y-intercept (0,b) is defined by
ymxb
=+.
(1.2)
The
point-slope form:
The line through the point (x
1
,y
1
) with slope m is defined
by
(
)
yy mxx
-=
-
1
.
(1.3)
1
The
two-point form:
The line through two distinct points (x
1
,y
1
) and (x
2
,y
2
) is
defined by
-=
-
-
yy
xx
xx
2
1
(
)
yy
-
1
.
(1.4)
1
2
1
Note that equations (1.2) and (1.3) above apply
only
to
nonvertical
lines.
When one wants to define lines in higher dimensions, then one can no longer use
a single equation and so we now give an alternative definition that works in all dimen-
