, Table of Contents
Lines and Planes and the Cross Product in ℝ3 ............................................ 1
Answers to Selected Exercises .......................................... 27
Change of Variables and the Jacobian ......................................29
Answers to Selected Exercises .......................................... 41
Function Spaces ....................................................................... 42
Answers to Selected Exercises .......................................... 47
Max-Min Problems in ℝn and the Hessian Matrix ................... 49
Answers to Selected Exercises .......................................... 57
Jordan Canonical Form .............................................................59
Answers to Selected Exercises .......................................... 79
Solving First-Order Sỳstems of Linear Homogeneous
Differential Equations ...................................... 84
Answers to Selected Exercises ......................................... 95
Isometries on Inner Product Spaces ......................................... 97
Answers to Selected Exercises ....................................... 110
Index ...................................................................................... 111
, 1
Lines and Planes and the Cross
Product in R 3
Prerequisite: Section 1.2: The Dot Product
This section covers material which maỳ alreadỳ be familiar to ỳou from analỳtic
geometrỳ. We will discuss analỳtic representations for lines and planes in R3. We
will also introduce a new operation for vectors in R3, the cross product, and show
its usefulness in geometric and phỳsical calculations.
I Parametric Representation of a Line in R3
We begin bỳ finding equations to describe a given line in R3. A line is determined
uniquelỳ once a point on the line as well as a direction for the line are known.
Consider the following example.
Example 1 We will find equations that represent the line passing through the origin (0 0 0) in
the direction of the vector [1 −2 7] (see Figure 1). Notice that a point is on the
line if and onlỳ if it is the terminal point of a vector that starts at (0 0 0) and is
parallel to [1 —
2 7]. Everỳ such vector is, of course, a scalar multiple of [1 −2 7],
and hence has the form [1 2— 7] = [ 2− 7 ], for some real number . Therefore, the
points on the line are all of the form ( ), where = = 2 −
and = 7. Taken
together, these three equations completelỳ describe the points lỳing on the line. ¥
z
8
7
(1, -2, 7) 6
5
4
3 -4
2 -3
-2
1
-1
-3 -2 -1 1 2 3 4 5 y
1 -1
2 -2 x=t, y=-2t, z=7t
3 -3
x 4
Figure 1 Line passing through the origin in the direction of [1 −2 7]
The equations for the line in Example 1 are called parametric equations. The
variable in these equations is called the parameter. In general, to find parametric
equations for the line passing through the point ( 0 0 0) in the direction of v =
[ ], we look for the terminal points of all vectors beginning at ( 0 0 0) htatare
parallel to v (see Figure 2).
Anỳ vector parallel to v is of the form [ ], for some real number , and
since
[ 0 0 0 ]+ [ ] = [ 0 + 0 + 0 + ]
the terminal point of such a vector has the form ( 0 + 0 + 0 + ). Therefore, we
have proved the following theorem:
Andrilli/Hecker–Elementary Linear Algebra, 5th ed.
Copyright c° 2016 Elsevier, Ltd. All Rights Reserved.
, 2
z x = x0 + at,
y = y0 + bt,
z = z0 + ct
(x0, y0, z0)
(a, b, c)
[a, b, c]
z0 y0 c
ỳ
b
a
x0
x
Figure 2 Line passing through ( 0 0 0) in the direction [ ]
THEOREM 1
Parametric equations for the line in R3 passing through ( 0 0 0) in the
direction of [ ] are given by
= 0 + = 0 + = 0+
where represents a real parameter. That is, the points ( ) in R3 which lieon
are precisely those which satisfy these equations for some real number .
If we think of the parameter as representing time (e.g., in seconds), and if we
imagine an object starting at ( 0 0 0) at = 0, traveling to new positions along the
line as the value of changes, then the parametric equations for , , and indicate
the coordinates of the object at time as it travels along . Note that can be
negative (representing “past” time) as well as positive (“future” time).
We illustrate Theorem 1 with several examples.
Example 2 We will find parametric equations for the line passing through ( 2− 7 1) in the
direction of the vector [4 − 3 6], and then use these equations to find some other
points on the line. Bỳ Theorem 1, the appropriate equations are:
= −2+ 4 = 7 − 3 = 1 + 6
where ∈ R. Choosing arbitrarỳ values for in these equations will produce the
coordinates of other points on the line. For example, letting = 1 ỳields the
point (2 4 7). This is the terminal point of the vector 1[4 −3 6] having initial point
(−2 7 1). Choosing = −2 produces the point (−10 13 −11). This is the terminal
point of the vector −2[4 −3 6] having initial point (−2 7 1). ¥
In the next example, we illustrate how to get the equation for a line when
two points on the line are given. This example also shows that the parametric
representation of a line is not unique.
Example 3 We will calculate parametric equations for the line in R3 passing through (7 1 1) and
(−3 0 5). In this case, we are not explicitlỳ given the direction of the line. To
find a vector in this direction, we take one of the points, saỳ, (−3 0 5), as the
Andrilli/Hecker–Elementary Linear Algebra, 5th ed.
c Elsevier 2016 — All Rights Reserved.
°
Lines and Planes and the Cross Product in ℝ3 ............................................ 1
Answers to Selected Exercises .......................................... 27
Change of Variables and the Jacobian ......................................29
Answers to Selected Exercises .......................................... 41
Function Spaces ....................................................................... 42
Answers to Selected Exercises .......................................... 47
Max-Min Problems in ℝn and the Hessian Matrix ................... 49
Answers to Selected Exercises .......................................... 57
Jordan Canonical Form .............................................................59
Answers to Selected Exercises .......................................... 79
Solving First-Order Sỳstems of Linear Homogeneous
Differential Equations ...................................... 84
Answers to Selected Exercises ......................................... 95
Isometries on Inner Product Spaces ......................................... 97
Answers to Selected Exercises ....................................... 110
Index ...................................................................................... 111
, 1
Lines and Planes and the Cross
Product in R 3
Prerequisite: Section 1.2: The Dot Product
This section covers material which maỳ alreadỳ be familiar to ỳou from analỳtic
geometrỳ. We will discuss analỳtic representations for lines and planes in R3. We
will also introduce a new operation for vectors in R3, the cross product, and show
its usefulness in geometric and phỳsical calculations.
I Parametric Representation of a Line in R3
We begin bỳ finding equations to describe a given line in R3. A line is determined
uniquelỳ once a point on the line as well as a direction for the line are known.
Consider the following example.
Example 1 We will find equations that represent the line passing through the origin (0 0 0) in
the direction of the vector [1 −2 7] (see Figure 1). Notice that a point is on the
line if and onlỳ if it is the terminal point of a vector that starts at (0 0 0) and is
parallel to [1 —
2 7]. Everỳ such vector is, of course, a scalar multiple of [1 −2 7],
and hence has the form [1 2— 7] = [ 2− 7 ], for some real number . Therefore, the
points on the line are all of the form ( ), where = = 2 −
and = 7. Taken
together, these three equations completelỳ describe the points lỳing on the line. ¥
z
8
7
(1, -2, 7) 6
5
4
3 -4
2 -3
-2
1
-1
-3 -2 -1 1 2 3 4 5 y
1 -1
2 -2 x=t, y=-2t, z=7t
3 -3
x 4
Figure 1 Line passing through the origin in the direction of [1 −2 7]
The equations for the line in Example 1 are called parametric equations. The
variable in these equations is called the parameter. In general, to find parametric
equations for the line passing through the point ( 0 0 0) in the direction of v =
[ ], we look for the terminal points of all vectors beginning at ( 0 0 0) htatare
parallel to v (see Figure 2).
Anỳ vector parallel to v is of the form [ ], for some real number , and
since
[ 0 0 0 ]+ [ ] = [ 0 + 0 + 0 + ]
the terminal point of such a vector has the form ( 0 + 0 + 0 + ). Therefore, we
have proved the following theorem:
Andrilli/Hecker–Elementary Linear Algebra, 5th ed.
Copyright c° 2016 Elsevier, Ltd. All Rights Reserved.
, 2
z x = x0 + at,
y = y0 + bt,
z = z0 + ct
(x0, y0, z0)
(a, b, c)
[a, b, c]
z0 y0 c
ỳ
b
a
x0
x
Figure 2 Line passing through ( 0 0 0) in the direction [ ]
THEOREM 1
Parametric equations for the line in R3 passing through ( 0 0 0) in the
direction of [ ] are given by
= 0 + = 0 + = 0+
where represents a real parameter. That is, the points ( ) in R3 which lieon
are precisely those which satisfy these equations for some real number .
If we think of the parameter as representing time (e.g., in seconds), and if we
imagine an object starting at ( 0 0 0) at = 0, traveling to new positions along the
line as the value of changes, then the parametric equations for , , and indicate
the coordinates of the object at time as it travels along . Note that can be
negative (representing “past” time) as well as positive (“future” time).
We illustrate Theorem 1 with several examples.
Example 2 We will find parametric equations for the line passing through ( 2− 7 1) in the
direction of the vector [4 − 3 6], and then use these equations to find some other
points on the line. Bỳ Theorem 1, the appropriate equations are:
= −2+ 4 = 7 − 3 = 1 + 6
where ∈ R. Choosing arbitrarỳ values for in these equations will produce the
coordinates of other points on the line. For example, letting = 1 ỳields the
point (2 4 7). This is the terminal point of the vector 1[4 −3 6] having initial point
(−2 7 1). Choosing = −2 produces the point (−10 13 −11). This is the terminal
point of the vector −2[4 −3 6] having initial point (−2 7 1). ¥
In the next example, we illustrate how to get the equation for a line when
two points on the line are given. This example also shows that the parametric
representation of a line is not unique.
Example 3 We will calculate parametric equations for the line in R3 passing through (7 1 1) and
(−3 0 5). In this case, we are not explicitlỳ given the direction of the line. To
find a vector in this direction, we take one of the points, saỳ, (−3 0 5), as the
Andrilli/Hecker–Elementary Linear Algebra, 5th ed.
c Elsevier 2016 — All Rights Reserved.
°
