1.Let u=(−2,0,4), v=(3,−1,6), and w=(2,−5,−5). Compute
(a) 3v−2u (b)u+v+w
(c) the distance between−3uandv+5w (d) projwu (e) u·(v×w) (f ) (−5v+w)×((u·v)w)
2.Repeat Exercise 1 for the vectorsu=3i−5j+k, v= −2i+2k, andw= −j+4k.
3.Repeat parts (a)–(d) of Exercise 1 for the vectors u=(−2,6,2,1),v=(−3,0,8,0), and w=(9,1,−6,−6).
4.(a) The set of all vectors inR2that are orthogonal to a nonzero vector is what kind of geometric object?
(b) The set of all vectors inR3that are orthogonal to a nonzero vector is what kind of geometric object?
(c) The set of all vectors inR2 that are orthogonal to two noncollinear vectors is what kind of geometric object?
(d) The set of all vectors inR3 that are orthogonal to two noncollinear vectors is what kind of geometric object?
5.LetA, B, andC be three distinct noncollinear points in 3- space. Describe the set of all pointsP that satisfy the vector equation−→
APⴢ(−→
AB×−→
AC)=0.
6.LetA, B, C,and Dbe four distinct noncollinear points in 3-space. If−→
AB×−→
CD=0and−→
ACⴢ(−→
AB×−→
CD)=0, explain why the line throughAandBmust intersect the line through CandD.
7.Consider the pointsP (3,−1,4),Q(6,0,2), andR(5,1,1). Find the pointSinR3whose first component is−1 and such that−→
PQis parallel to−→
RS.
8.Consider the points P (−3,1,0,6), Q(0,5,1,−2), and R(−4,1,4,0). Find the pointS inR4 whose third compo- nent is 6 and such that−→
PQis parallel to−→
RS.
9.Using the points in Exercise 7, find the cosine of the angle between the vectors−→
PQand−→
PR.
10.Using the points in Exercise 8, find the cosine of the angle between the vectors−→
PQand−→
PR.
11.Find the distance between the pointP (−3,1,3)and the plane 5x+z=3y−4.
12.Show that the planes 3x−y+6z=7 and
−6x+2y−12z=1 are parallel, and find the distance be- tween them.
In Exercises13–18, find vector and parametric equations for the line or plane in question.
13.The plane in R3 that contains the points P (−2,1,3), Q(−1,−1,1), andR(3,0,−2).
14.The line inR3that contains the pointP (−1,6,0)and is or- thogonal to the plane 4x−z=5.
15.The line inR2that is parallel to the vectorv=(8,−1)and contains the pointP (0,−3).
16.The plane in R3that contains the pointP (−2,1,0)and is parallel to the plane−8x+6y−z=4.
17.The line inR2with equationy=3x−5.
18.The plane inR3with equation 2x−6y+3z=5.
In Exercises19–21, find a point-normal equation for the given plane.
19.The plane that is represented by the vector equation (x, y, z)=(−1,5,6)+t1(0,−1,3)+t2(2,−1,0).
20.The plane that contains the pointP (−5,1,0)and is orthogo- nal to the line with parametric equationsx=3−5t,y=2t, andz=7.
21.The plane that passes through the pointsP (9,0,4), Q(−1,4,3), andR(0,6,−2).
22.Suppose thatV = {v1,v2,v3}andW= {w1,w2}are two sets of vectors such that each vector inVis orthogonal to each vec- tor inW. Prove that ifa1, a2, a3, b1, b2are any scalars, then the vectorsv=a1v1+a2v2+a3v3andw=b1w1+b2w2are orthogonal.
23.Show that in 3-space the distancedfrom a pointP to the line Lthrough pointsAandBcan be expressed as
d= −→
AP×−→
AB −→
AB
24.Prove thatu+v = u + vif and only if one of the vec- tors is a scalar multiple of the other.
25.The equationAx+By=0 represents a line through the ori- gin inR2ifAandBare not both zero. What does this equation represent inR3if you think of it asAx+By+0z=0? Ex- plain.
183
C H A P T E R 4
General Vector Spaces
CHAPTER CONTENTS 4.1 Real Vector Spaces 183 4.2 Subspaces 191
4.3 Linear Independence 202 4.4 Coordinates and Basis 212 4.5 Dimension 221
4.6 Change of Basis 229
4.7 Row Space, Column Space, and Null Space 237 4.8 Rank, Nullity, and the Fundamental Matrix Spaces 248 4.9 Basic Matrix Transformations inR2andR3 259 4.10 Properties of Matrix Transformations 270 4.11 Geometry of Matrix Operators onR2 280
INTRODUCTION Recall that we began our study of vectors by viewing them as directed line segments (arrows). We then extended this idea by introducing rectangular coordinate systems, which enabled us to view vectors as ordered pairs and ordered triples of real numbers.
As we developed properties of these vectors we noticed patterns in various formulas that enabled us to extend the notion of a vector to ann-tuple of real numbers.
Althoughn-tuples took us outside the realm of our “visual experience,” it gave us a valuable tool for understanding and studying systems of linear equations. In this chapter we will extend the concept of a vector yet again by using the most important algebraic properties of vectors inRnas axioms. These axioms, if satisfied by a set of objects, will enable us to think of those objects as vectors.
4.1 Real Vector Spaces
In this section we will extend the concept of a vector by using the basic properties of vectors inRnas axioms, which if satisfied by a set of objects, guarantee that those objects behave like familiar vectors.
Vector Space Axioms The following definition consists of ten axioms, eight of which are properties of vectors inRnthat were stated in Theorem 3.1.1. It is important to keep in mind that one does notproveaxioms; rather, they are assumptions that serve as the starting point for proving theorems.
DEFINITION1 LetV be an arbitrary nonempty set of objects on which two operations are defined: addition, and multiplication by numbers calledscalars. Byadditionwe mean a rule for associating with each pair of objectsuandvinV an objectu+v, called thesumofuandv; byscalar multiplicationwe mean a rule for associating with each scalarkand each objectuinV an objectku, called thescalar multipleofubyk.
If the following axioms are satisfied by all objectsu,v,winV and all scalarskand m, then we callV avector spaceand we call the objects inV vectors.
1. Ifuandvare objects inV,thenu+vis inV.
2. u+v=v+u
3. u+(v+w)=(u+v)+w
4. There is an object0inV,called azero vectorforV,such that0+u=u+0=u for alluinV.
5. For eachuinV,there is an object−uinV,called anegativeofu, such that u+(−u)=(−u)+u=0.
6. Ifkis any scalar anduis any object inV,thenkuis inV.
7. k(u+v)=ku+kv 8. (k+m)u=ku+mu 9. k(mu)=(km)(u) 10. 1u=u
In this text scalars will be ei- ther real numbers or complex numbers. Vector spaces with real scalars will be calledreal vector spaces and those with complex scalars will be called complex vector spaces. There is a more general notion of a vector space in which scalars can come from a mathematical structure known as a “field,”
but we will not be concerned with that level of generality.
For now, we will focus exclu- sively on real vector spaces, which we will refer to sim- ply as “vector spaces.” We will consider complex vector spaces later.
Observe that the definition of a vector space does not specify the nature of the vectors or the operations. Any kind of object can be a vector, and the operations of addition and scalar multiplication need not have any relationship to those on Rn. The only requirement is that the ten vector space axioms be satisfied. In the examples that follow we will use four basic steps to show that a set with two operations is a vector space.
To Show That a Set with Two Operations Is a Vector Space Step 1. Identify the setV of objects that will become vectors.
Step 2. Identify the addition and scalar multiplication operations onV.
Step 3. Verify Axioms 1 and 6; that is, adding two vectors inV produces a vector inV, and multiplying a vector inV by a scalar also produces a vector inV.
Axiom 1 is calledclosure under addition, and Axiom 6 is calledclosure under scalar multiplication.
Step 4. Confirm that Axioms 2, 3, 4, 5, 7, 8, 9, and 10 hold.
Hermann Günther Grassmann (1809–1877)
Historical Note The notion of an “abstract vector space” evolved over many years and had many contributors. The idea crystallized with the work of the German mathematician H. G. Grassmann, who published a paper in 1862 in which he con- sidered abstract systems of unspecified elements on which he defined formal operations of addi- tion and scalar multiplication. Grassmann’s work was controversial, and others, including Augustin Cauchy (p. 121), laid reasonable claim to the idea.
[Image: © Sueddeutsche Zeitung Photo/The Image Works]
4.1 Real Vector Spaces 185 Our first example is the simplest of all vector spaces in that it contains only one object. Since Axiom 4 requires that every vector space contain a zero vector, the object will have to be that vector.
E X A M P L E 1 The Zero Vector Space
LetV consist of a single object, which we denote by0, and define 0+0=0 and k0=0
for all scalarsk. It is easy to check that all the vector space axioms are satisfied. We call this thezero vector space.
Our second example is one of the most important of all vector spaces—the familiar spaceRn. It should not be surprising that the operations onRnsatisfy the vector space axioms because those axioms were based on known properties of operations onRn.
E X A M P L E 2 RnIs a Vector Space
LetV =Rn, and define the vector space operations onV to be the usual operations of addition and scalar multiplication ofn-tuples; that is,
u+v=(u1, u2, . . . , un)+(v1, v2, . . . , vn)=(u1+v1, u2+v2, . . . , un+vn) ku=(ku1, ku2, . . . , kun)
The setV =Rnis closed under addition and scalar multiplication because the foregoing operations producen-tuples as their end result, and these operations satisfy Axioms 2, 3, 4, 5, 7, 8, 9, and 10 by virtue of Theorem 3.1.1.
Our next example is a generalization ofRnin which we allow vectors to have infinitely many components.
E X A M P L E 3 The Vector Space of Infinite Sequences of Real Numbers LetV consist of objects of the form
u=(u1, u2, . . . , un, . . .)
in whichu1, u2, . . . , un, . . .is an infinite sequence of real numbers. We define two infi- nite sequences to beequalif their corresponding components are equal, and we define addition and scalar multiplication componentwise by
u+v=(u1, u2, . . . , un, . . .)+(v1, v2, . . . , vn, . . .)
=(u1+v1, u2+v2, . . . , un+vn, . . .) ku=(ku1, ku2, . . . , kun, . . .)
In the exercises we ask you to confirm thatV with these operations is a vector space. We will denote this vector space by the symbolR⬁.
Vector spaces of the type in Example 3 arise when a transmitted signal of indefinite
E(t) Voltage
Time t 1
–1
Figure 4.1.1
duration is digitized by sampling its values at discrete time intervals (Figure 4.1.1).
In the next example our vectors will be matrices. This may be a little confusing at first because matrices are composed of rows and columns, which are themselves vectors (row vectors and column vectors). However, from the vector space viewpoint we are not