Explain why C is a basis for H

Mathematics by Geoffrey
Linear Algebra and Differential Equations
4.5 EXERCIS ES

Fnr each , ul.J sp;1cc in EX L’rl’ iscs 1-8. (a) fi nd a basis, and (b) state
rli ~ dirn cnsilll1.
L l[ \ :,2,} ‘ 1 i” R} 2. { [ -ii] , I in R}
3. j[ ! [iJ a.b. c in R} 4. {[ :aI: l a. b inR l 5. { [ {a ~
4
; –/:, l : a, b , c in JR }
-3a + 7b + 6c
6. {[£~2
t{~cl : a ,b,c in JR }
– 3a + b + c
7. {(a. b, c ) : a – 3b + c = 0,b – 2c = 0, 2b – c = O}
8. {(a .h,c,d) : a – 3b + c = 0}
9. Fi nd th e di mensio n of the subspace of all vectors in JR 3 who se
firs t and third entries are equal.
10 . Fin d the dimension of the subspace H of JR 2 spanned by
[ –~ J [ ~~ J [ -! J In Ex erci ses 11 and 12 , fi nd the dimens ion of the subspace
spann ed by the given vectors .
JI. m-nH _n [ =n 12
• [-nnJf nffl Determin e th e dimensions of Nu ! A and Col A fo r the matr ice s
show n in Ex ercises l 3- I 8 .
[!
-6 9 0
-;] 13 . , t = I 2 – 4
0 0 5 I
0 0 0 0
I~ ll 3 -4 2 – I
-~l 14. 0 I – 3 7
0 0 I 4
0 0 0 0
15 . ,\ = [:) 0 9 _:] 0
16. I ·= r 3 I~] -6
4.5 The Dimension of a Vector Space 231
11. A= [~] -]
4
0 n 18. A= [~] 4
7
0
In Exercises 19 and 20, JI is a vector space. Mark each statement
True or Fal se. Justify each answ er.
19 . a. Th e numb er of pivot columns of a matrix equals th e
dimension of its co lumn space .
b. A plane in JR 3 is a two-dimen sional subspace of lll 3.
C . Th e dimension of th e vector spac e IF’4 is 4.
d. If dim V = n and S is a lin early independent set in JI,
then S is a basis for JI.
e. If a set { v 1, . .. , v P} spans a finite-dimensional vector
space V and if T is a set of more than p vectors in V,
th en T is linearly dependent.
20. a. JR 2 is a two-dimensional subspace of JR 3 .
b. The numb er of vari abl es in the equation Ax = 0 equals
the dimension of Nu! A.
C. A vec tor space is infinite-dimensional if it is spanned by
an infinite se t.
d . If dim JI = n and if S spans V, then S is a basis of V .
e. The onl y three-dimensional subspace of JR 3 is JR 3 itself.
21. The fir st four Hermite polynomi als are 1, 2t, – 2 + 4£ 2 , and
– I 2t + 8t 3 • These polynomials arise naturally in the study
of certai n important differential equations in mathematical
ph ys ics. 2 Show th at the fir st four Hermite polynomials form
a ba sis of IF’3 .
22. The fir st four Laguerre pol ynomials are 1, I – t, 2 – 41 + 12 ,
and 6 – I 81 + 912 – t 3 . Show that these polynomials fom1 a
bas is of IF’3.
23 . Le t B be the bas is of IF’3 con sisting of the Hermite po ly no-
mials in Exerci se 21, and let p (t) = 7 -12t – 8t 2 + 121 3 .
Find the coordinate vector of p relative to B.
24 . Let B be th e basis of IF’2 con sisti ng of the fir st three
Lagu erre polynomials listed in Exercise 22, and let
p (l) = 7 – 81 + 31 2 • F ind the coordinate vector of p rel ativ e
toB .
25 . Let S be a sub set of an n -dimensional vector space JI . and
sup pose S contains few er than n vect ors. Explain why S
ca nnot sp an V.
26 . Let H be an 11-climcnsional subspace of an n -dim ensional
vec to r sp ace V . Show that H = JI .
27 . Ex pla in why the space 1P’ of all pol ynomial s is an infinit e-
di me nsion al spac e.
2 See l 111 rod11 cti o 11 tu F11 11c ti o1w/ A11 a/_l’l’i s , 2nd ed .. by A . E. T;1 y lor ancl
David C . Lay (New Yo rk: John Wil ey & So ns, 198 0), pp . 92-93. Ot her
,e ls of po lynomi al~ arc di ~c u~sc cl th ere. 10 0 .
\

232 CHAPTER 4 Vector Sp acl~.s
28. Show that th-: spJc~ C UI~ ) of a~l cont~nuous functions defi ned
1..m the real Jin~ is an i111111ite-d11nens1onal space.
In Exncis..: s :29 .md 30, V is a nonzero finite-dimensional vector
“‘l ~,, ,111d the \’ ~cto rs listed belong to V . Mark each statement Sp. L~ , • .
True or Fulse. Ju stify each answer. (These questions are more
di f;ic ul t th:m those in Exercis es 19 and 20.)
29. a . If the re exists a set { V1, … , v p} that spans V, then
dim l’ ::: p.
b. If there exists a linearly independent set { v 1, . .. , v P} in
V , then dim V > p.
c . If dim V = p, then there exists a spanning set of p + I
vectors in V.
30. a. If there exists a linearly dependent set { v 1, .. . , v P} in V,
then dim V < p .
b. If every set of p elements in V fails to span V, then
dim V > p .
c . If p > 2 and dim V = p , then every set of p – I nonzero
vectors is linearly independent.
Exercises 31 and 32 concern finite-dimensional vector spaces V
and W and a linear transformation T : V -+ W .
31. Let H be a nonzero subspace of V, and let T (H) be the set of
images of vectors in H. Then T(H) is a subspace of_ W, by
Exercise 35 in Section 4 .2 . Prove that dim T(H) < dim H.
32. Let H be a nonzero subspace of V, and suppose T is
a one-to-one (linear) mapping of V into W . Prove that
dim T (H) = dim H. If T happens to be a one-to-o~e m~p-
ping of V onto W, then dim V = di m W . I~omo~pluc fimte-
dimensional vector spaces have the same d1mens10n.
33.
34.
11 a linearly independent set [M] According to Theorem a~ded to a basis for JR n. One } . JR ll can be e xp { v’ , … , v k m A – [ v I . . . v k e I . . . en ] ‘ · · t create –way to do this 1s O
f the identity matrix ; the pivot . the c olumns o w1the1, •• ·• e 11 . f IR?.11 f A form a basis or .
columns O
‘b d to extend the follow ing vectors Use the method descn e a . 5
to a basis for IR?. :
VJ=
– 9
-7
8
-5
7
9
4
6
-7
6
7
-8
5
– 7
Ex lain why the method works in general: ‘Y’hy are the
b. .P. . v included in the basis found for ongmal vect01 s v 1 , • •• , k
Col A ? Why is Col A = R 11 ?
[M] Let B = { 1 , cos t , cos2 t, … ‘ cos6 t} and C. = { 1, cos!’
cos 2t, . . . , cos 6t} . Assume the following tngonometnc
identities (see Exercise 37 in Section 4 .1) .
cos 2t = -1 + 2 cos2 t
cos 3t = -3 cost+ 4cos3 t
cos 4t = 1 – 8 cos2 t + 8 cos4 t
cos St = 5 cos t – 20 cos3 t + 16 cos5 t
cos 6t = -1 + 18 cos2 t – 48 cos4 t + 32 cos6 t
Let H be the subspace of functions spanned by the functions
in B. Then Bis a basis for H, by Exercise 38 in Section 4.3.
a. Write the B-coordinate vectors of the vectors in C, and
use them to show that C is a linearly independent set in H.
b. Explain why C is a basis for H .

3. vVhat is the next step to pcnu1~11 l,V …. ~ •- –helon form of AT? . , 1n a row ec 4. How many pivot columns dfC
4.6 E}{ERC~SES — 6 2 5 2 -3 In Exercises 1-4, assume that the matrix A is row equiv alent to B. -2 3 -3 -3 -4 Without calculations. list rank A and dim N ul A . Then fi nd bases 3. A == 9 5 9 ‘ 4 -6 for C ol A, Row A~ and Nul A. – 2 3 3 – 4 1
A=[-: -4 9
-7] 2 -3 6 2 5 1. 2 -4 1 ‘ 0 0 3 -1 1 5 -6 10 7 B == 0 0 0 1 3
B = [~
0
-~] 0 0 0 0 0 – 1
– 2 5
1 1 – 3 7 9 – 9 0 0
1 2 – 4 10 13 – 12 1 -3 4 – 1 9 4. A == 1 -1 -1 1 1 – 3 – 2 6 – 6 -1 – 10 1 – .3 1 – 5 -7 3 2 . A = 9 -6 – 6 – 3 ‘ 1 – 2 0 0 -5 -4 -3
3 – 9 4 9 0 1 1 -3 7 9 – 9 1 – 3 0 5 -7 0 1 – 1 3 4 -3 0 2 -3 8 B= 0 0 0 1 -1 – 2 0 B= 0 0 0 5 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0

– If a ~ x 8 matrix A has rank 3 fi nd er . N 1 :,. J .. k ;lr ‘ 1lll u ,1.d1 111RowA . an 1an .
(J If a 6 x 3 matrix A has ra nk J fi nd dii N 1 1 1 1′ k 1r · 11 u : .t1111Row 1, and r.in , .
7 Suppose a 4 x 7 matri x •t has r1111 r 1, i\’ ,t I I·. • “” “‘) . 1 · •\ 1:1 1 u11111 s . , Col A = .~ . ls Nul : t =- ll-R ‘! h .pln in )l’lll’ , 111:–,wrs .
S. SttPi’0″ 12 a :=; >.. (, matri\ I has t’l.l 1t r pivnt rnlu mns . W hat is di m Nul. I: ls Cnl. t = lk~ 1•.1 W hy nr wh y not ?
9, lf the null :–p,11..’ 1..’ tlf a ~ >.. (1 matrix A is 4-dimc nsi onal. wha t is the di1111..· 11s io11 (l f the co lumn space of ; I?
10. lf the nu ll sp ,h:c of a 7 x 6 matri x A is 5-dimensional, what is 111c dimensi on of the colu mn space of A?
11. lf th e null 11pace of an 8 x 5 matrix A is 2-di mensional , what is the dimension of the row space of A?
1~. If the null space of a 5 x 6 matrix A is 4-di mensional , what i, the dimension of the row space of A?
1.3. If .4 is a 7 x 5 matrix, what is the largest possible rank of A?
!f rl is a 5 x 7 matrix, wh at is the largest possible rank of A? Exp lain your answers.
l .t . If A is a 4 x 3 matrix, wh at is the largest possible dimension of the row space of A? If A is a 3 x 4 matrix, what is the largest possible dimension of the row sp ace of A? Explain.
15. If .4 is a 6 x 8 matri x , what is the smallest possible dimension of Nul A?
16. If A is a 6 x 4 matri x, what is the smallest possible dimension
of Nul A?
: 1 Exercise s 17 and 18, A is an m x n matri x . Mark each statement
True or False. Justify each an swer.
l 7. a. The row space of A is the same as the column space of
AT.
18.
b. If B is any echelo n form of A , an d if B has three nonzero
rows then the first three rows of A fonn a bas is for ‘
C.
d.
e.
a.
b.
C.
d.
Row A .
The dimensions of the row space and the column space
of A are the same , even if A is not square .
The <, um of the dimensions of the ro w space and the null
space of A equals the number of rows in A.
On a computer, row operations can change the app are nt
rank of a matrix.
If B is any echelon form of A, th en th e pivot columns of
B form a basis for the co lum n -, pace of A.
Row operations preserve the linear de pend ence relatio ns
among the ro ws of A. .
The dimension or the null !-. pace of ,1 is the number ol
col umns of A that arc not pivot colum ns.
The row space of Ar is the same as the column space of
A.
4.6 Rank 239
c. lf 1\ and B are row equ ivalent, the n their row spaces arc the ,amc .
19 . ~upposc the solutions of u homo gen eou s sys tem of five li ne ar c411atio n.., ll1 six un kno wns arc all mult iples of one no nze ro ,olutio11. Wi ll the sy -.t cm nece ssaril y have a sol ution for cwry pm,,ih lc choice o f co nsta nt s on the right side s of the equation .., ‘! Lxpl am.
20. Sup pose a nonhomogcncous sy~tc m of ‘>i x lin ear equ at io ns in ei ght unknow ns ha,; a ‘>o lut ion, with two fr ee variables. Is it possible to chan ge some constants on the equ ations ‘ right si des to make the new sys tem inco nsb,tc nt: Explain.
21. Suppose a no nho mo geneou s system of ni ne linear eq uations in ten unkn owns has a sol uti on for all possible cons tants on the righ t side s of the equ atio ns. Is it possible to fi nd two non zero so lutions of the ass ociated homogeneous system that are not multiples of each other: Discuss .
22. Is it possible that al l solut io ns of a ho mogeneo us system of ten linear eq uations in twe lve variables are multiples of one fixed non zero solution? Di scus s.
23. A homogeneous system of twelve li near eq uations in eight unknowns ha s two fixed so lutions that are no t multiples of each other, and all other sol utions are linear combinati ons of these two solutions. Can the set of al l solutions be de scribed with fewer than twelve homogeneous lin ear equations? If so. how many? Discuss.
24. Is it possible for a nonhomogeneous syste m of seven eq ua -tions in six unknowns to have a unique solution for some right-hand side of constants? Is it possible for such a system to have a unique solution for every right-hand side? Explain.
25 . A scientist solves a nonhomogeneous system of ten linear equations in twelve unknow ns an d fi nds that three of the unknowns are free variables. Can the scientist be certain that, if the right side s of the equatio ns are changed, the new nonhomogeneous system will have a sol ution? Disc uss .
26. In statistical theory, a common requirement is that a matrix be of full ra nk. That is, the rank should be as large as
possi ble. Explain why an m x n matrix with more rows than
columns has full rank if and only if its columns are linearly
independent.
Exercises 27-29 concern an m x n matrix A :md what are often
called the fun da menta l subspaces detennined by A .
27 .
28.
29.
T Which of the su bspaces Row .4. Co l A. Nu t A, Row A , Col AT. an d Nul AT are in !Rm an d whi ch are in lR. 11 ? How
many distinct subspaces arc in this list?
Justify th e fo ll o\\’ing equali ties:
a . Jim Row A + dim Nu! A = 11 ‘- tnnb , r o f L·o lunms or A
b. dim Col A+ d im Nu ! AT = m Numb er of rows of : \
Use faercise 28 to explain why the equation Ax = b has a so lution for all b in ~ 111 if and only if the equation AT x = 0
has only the trivial solution.

240 CHAPTER 4 \ cctor Sp.tees
30. Suppm,e , t ism ‘- 11 ~md bis in ~ 111 • What has to be tru e about
the t,\·o numb~rs rnnk l A b ] and rank A in order for the
cquJti(“‘tl Ax = b to be consistent?
R:.mk I matri~es are important in some computer algorithms and
se\ cral theoretical contexts , including the singular value decom-
l~osition in Chapter 7. It can be shown that an ,n x n matrix A
hJ~ r:nk 1 if and only if it is an outer product; that is, A = uvr
for some _u in lRm and v in ~ 11 • Exercises 31-33 suggest why this
property 1s true.
31. Verify th at rank uv7
< I if u = [ -n and v = [fl·
32. Letu = [; l Find v in R. 3 such that U =! : ] = uvr.
33. Let A be any 2 x 3 matrix such that rank A = 1, let u be the
first col umn of A, and suppose u # 0. Explain why there
is a vector v in JR 3 such that A = uvr. How could this
construction be modifi ed if the first column of A were zero?
34. Let A be an m x n matrix of rank r > 0 and let U be an ech-
elon form of A . Explain why there exists an invertible matrix
E such that A = EU , and use this factorization to write A
as the sum of r rank 1 matrices . [Hin t: See Theorem 10 in
Section 2.4 .]
7 -9 -4 5 3 -3 -7
-4 6 7 – 2 -6 -5 5
A 5 _ 7 -6 5 -6 2 8
35. [M] Let = 7 -3 5 8 -1 – -4 8
36.
37.
38.
6 -8 -5 4 4 9 3
Construct matrices C and N whose columns are bases for ,
a. Col A and Nul A, respectively, and construct a matrix R
whose rows form a basis for Row A.
b. Construct a matrix M whose column[sRtorm a ba.
sis for Nul Ar, form the matrices S = N ] and
T = [ c M ] , and explain why S and T should be
square. Verify that both S and T are invertible.
[M] Repeat Exercise 35 for a random integer-valued 6 x 7
matrix A whose rank is at most 4 . One way to make A
is to create a random integer-valued 6 x 4 matrix J and a
random integer-valued 4 x 7 matlix K, and set A = J K.
(See Supplementary Exercise 12 at the end of the chapter;
and see the Study Guide for matrix-generating programs.)
[M ] Let A be the matrix in Exercise 35. Construct a matrix
C whose columns are the pivot columns of A, and construct
a matrix R whose rows are the nonzero rows of the reduced
echelon fo rm of A . Compute CR, and discuss what you see.
[M] Repeat Exercise 37 for three random integer-valued
? x 7 matrices A whose ranks are 5, 4 , and 3. Make a con-
Jecture about how CR is related to A ~ . A p . 1or any matnx . rove your conJecture.

4 . 7 EX ERCISES
1. Let B = {b 1, b2 } and C = {c 1, c2 } be bases for a vector space
V , and suppose b 1 = 6c1 – 2c2 and b2 = 9c1 – 4c2.
a . Find the change-of-coordinates matrix from B to C.
b. Find [ x 1c for x = – 3b 1 + 2b2 . Use part (a).
2. Let B = {b 1, h2} and C = { c 1, c2 } be bases for a vector space
V, and suppose b 1 = -c1 + 4c2 and b2 = Sc 1 – 3c2 .
a. Find the change-of-coordinates matrix from B to C.
b. Find [ x 1c for x = Sb1 + 3b2 •

\ L,: t U == {u 1, U2} and W = {w 1 w } b b · · · · 2 e ases for V and let p be a matnx whose columns are [u ] [ ‘ . • 1 · . . 1 w nncl Uz ] w . Wluch of th e fol owmg equat10ns 1s satisfied by p f 11 . or a x 111 V?
(i) [x]u = P[xlw (ii) [ x)w = P [ x]u
4. Let A= {a 1, a2,a3 } and D = {d1.d2,d3} be bases fo r V,
and let P_ = [ _[di)~ _ [d2)..4 (d3]A ] . W hich of the fo ll ow-in g equat t0ns 1s sat1shcd by P for nll x in V’?
(i ) [x]A = P[x]v (ii) [ x )v = P [ x ]A
5. Let A={a 1, a 2.a., } and B ={b 1,b2, b 3 } be bases
for a ve ctor space V, and suppose a 1 = 4b 1 _ bz,
a2 = -b1 + b2 + b3.anda3 = b 2 -2b3.
a. Find the change-of-coordinates m atrix from A to B.
b. Find [x] 13 forx = 3a1 + 4a2 + a 3 •
6. Let V = {d 1. d2, d3} and F = {f1, f2 , f3 } be bases for
a vector space V, and suppose f 1 = 2d1 _dz + d3 ,
f2 = 3d 2 + d3, and f3 = – 3d 1 + 2d3 .
a. F in d the change-of-coordinates matrix from F to D.
b. Find [ x lv for x = f1 – 2f2 + 2f3 .
In Exercises 7-10, let B = {b1 , b 2} and C = {c 1, c2 } be bases for
3 2. In each exercise, fi nd the change-of-coordinates matrix from l3 to C and the change-of-coordinates matri x from C to B.
7. b 1 = [~].b2 = [ =n ,c1 = [ _;J. c2 = [-~ J
8. b 1 = [ – ~ l b2 = [ _; l C1 = [ ! l Cz = [ ~]
9. b 1 = [ =n ,b2 =[~l ei= [-ilc2 = [-~]
10. b1 = [ _;J.b2 = [-ilc1 = Ulc2 = [ ; ]
In Ex ercises 11 and 12, B and C are bases for a vec tor space V . \ti. ark each statement True or False. Justify each answer.
11. a. The columns of the change-of-coordinates matrix e!B are B-coordinate vectors of the vectors in C.
b. If V = ]Rn and C is the standard basis for V, then els is the same as the change-o f-coordinates matrix PB intro-
d uced in Section 4 .4.
12. a. The columns of els are linearly independent.
b. If V = JR 2, B = {b 1, b2}, and C = {c1,c2}, then row
reduction of [ c 1 c2 b 1 b2 ] to [ I P ] produces a
ma trix P that satisfies [ x ]13 = P [ x 1e for all x in V.
13. In 1P’2 , fin d the change-of-coordi nates matrix from the basis
B = {1 – 2t + t 2 , 3 – St+ 4t 2 , 2t + 3t 2} to the standard
bas is C = { 1, t . t 2 }. Then find the B-coordinate vector for
-] + 2t.
l 4, In lF’2 , fi nd the change-of-coordinates matrix from the ba-
sis B = { 1 – 3t 2 , 2 + t – 5t 2 , 1 + 2t} to the standard basis.
Then write t 2 as a linear combination of the polynomials in B.
4. 7 Change of Basis 245
Exerci ses 15 and 16 provide a proof of Theore m 15 . Fill in a
justifi cation for each step.
15 . Gi ven v in V, th ere exist scalars x1, ••• , x 11 , suc h that
because (a) __ . Apply the coordinate mapping deter-
min ed by th e basis C, and obtain
because (b) __ . This equation may be writte n in the form
(8)
by the definition of (c) _ _ . This sho ws that the matrix
elB shown in (5) satisfies [v]e = ef13 [v)13 for each v in V,
because the vector on the right side of (8) is (d) __ .
16. Suppose Q is any matrix such that
[v]c = Q[v)B for each v in V (9)
Set v = b I in (9) . Then (9) shows that (b 1] e is the first column of Q because (a) __ . Similarly, fork = 2, . . . , n, the kth col umn of Q is (b) __ because (c) _ _ . This shows
that the matrix clt3 defined by (5) in Theorem 15 is the only
matrix that satisfies condition (4).
17. [M] LetB = {xo, … ,x6}andC = {y0 , .. . ,y6},where xkis
the function cosk t and y k is the function cos kt. Exercise 34
in Section 4.5 showed that both B and C are bases for the
vector space H = Span {x 0 , .. . , x6}.
a . SetP =[[y0 ] 6 ( y GJ 6 ] ,andcalculateP-1.
b . Ex plain why the columns of p- 1 are the C-coordinate
vectors of Xo, . .. , x6 . Then use these coordinate vectors
to write trigonometric identities that express powers of
cost in terms of the functions in C.
See the Study Guide.
18. [M] (Calcu lus require d)3 Recall from calculus that inteorals b
such as
f (5cos3 t – 6cos4 t + 5cos5 t – 12cos6 t) dt (IO)
are tediou s to compute. (The usu al method is to apply inte-
gration by parts repeatedl y and use the half-angle formula .)
U se the matrix P or p – t from Exercis e 17 to transform (I O);
then compute the integral.
3 The idea for Exercises 17 and I 8 and five re lated exercises in earlier sections ca me from a p aper by Jack W. Rogers , Jr., of Auburn Uni versity, presented at a meet in g of th e Internation a l L inear Algebra Socie ty, Augu st 1995. See “Applications of Linear Algebra in Ca lculus.” American Mathematical Monthly 104 ( I), 1997.
1

246 CHAPTER 4 Vector Spaces
19. [~i] Lt>l
2
-5
6
-l] 0 )
J
a. Fi nd a basis {u 1• u2 , u3 } for Jffi. 3 such that P is the
cham!e-of-coordinates matrix from {u 1, u2, u3} to the
basis-{v 1• v2 . v3 } . [Hint: What do the columns of clB
:-epre sent ?]
20 .
b. Find a bas is {w 1, w2, w d for fl~ 1 wch tl”at p i ~ th(; cha n.;e –
of-coordin atcs matri x from { v 1, v2 , v ~} to f w. , ; r_; · w ~i}.
Let B = {b 1 b2 f1 C = fc c 1 ·111cl V – 1d d- I· b1..: be;.·.c.., ‘ ‘ I/• 2f, ‘ – l • ‘ –
for a two-d imens ional vector space .
W
. . , p p a. nte an equati on tha t rel ates th e ma tnCC ‘.) c – :3 • :;:::, -.: ·
and vf-B. Just ify you r res ul t.
b. [M] Use a matrix program eith er to help ~ ou f. nd rr- e
equ atio n or to check the equation you wTit c. \\”or~: \~ ith
three bases for JR 2 . (Se e Exe rci ses 7-1 0.)

4.5 EXERCIS ES
Fnr each , ul.J sp;1cc in EX L’rl’ iscs 1-8. (a) fi nd a basis, and (b) state
rli ~ dirn cnsilll1.
L l[ \ :,2,} ‘ 1 i” R} 2. { [ -ii] , I in R}
3. j[ ! [iJ a.b. c in R} 4. {[ :aI: l a. b inR l 5. { [ {a ~
4
; –/:, l : a, b , c in JR }
-3a + 7b + 6c
6. {[£~2
t{~cl : a ,b,c in JR }
– 3a + b + c
7. {(a. b, c ) : a – 3b + c = 0,b – 2c = 0, 2b – c = O}
8. {(a .h,c,d) : a – 3b + c = 0}
9. Fi nd th e di mensio n of the subspace of all vectors in JR 3 who se
firs t and third entries are equal.
10 . Fin d the dimension of the subspace H of JR 2 spanned by
[ –~ J [ ~~ J [ -! J In Ex erci ses 11 and 12 , fi nd the dimens ion of the subspace
spann ed by the given vectors .
JI. m-nH _n [ =n 12
• [-nnJf nffl Determin e th e dimensions of Nu ! A and Col A fo r the matr ice s
show n in Ex ercises l 3- I 8 .
[!
-6 9 0
-;] 13 . , t = I 2 – 4
0 0 5 I
0 0 0 0
I~ ll 3 -4 2 – I
-~l 14. 0 I – 3 7
0 0 I 4
0 0 0 0
15 . ,\ = [:) 0 9 _:] 0
16. I ·= r 3 I~] -6
4.5 The Dimension of a Vector Space 231
11. A= [~] -]
4
0 n 18. A= [~] 4
7
0
In Exercises 19 and 20, JI is a vector space. Mark each statement
True or Fal se. Justify each answ er.
19 . a. Th e numb er of pivot columns of a matrix equals th e
dimension of its co lumn space .
b. A plane in JR 3 is a two-dimen sional subspace of lll 3.
C . Th e dimension of th e vector spac e IF’4 is 4.
d. If dim V = n and S is a lin early independent set in JI,
then S is a basis for JI.
e. If a set { v 1, . .. , v P} spans a finite-dimensional vector
space V and if T is a set of more than p vectors in V,
th en T is linearly dependent.
20. a. JR 2 is a two-dimensional subspace of JR 3 .
b. The numb er of vari abl es in the equation Ax = 0 equals
the dimension of Nu! A.
C. A vec tor space is infinite-dimensional if it is spanned by
an infinite se t.
d . If dim JI = n and if S spans V, then S is a basis of V .
e. The onl y three-dimensional subspace of JR 3 is JR 3 itself.
21. The fir st four Hermite polynomi als are 1, 2t, – 2 + 4£ 2 , and
– I 2t + 8t 3 • These polynomials arise naturally in the study
of certai n important differential equations in mathematical
ph ys ics. 2 Show th at the fir st four Hermite polynomials form
a ba sis of IF’3 .
22. The fir st four Laguerre pol ynomials are 1, I – t, 2 – 41 + 12 ,
and 6 – I 81 + 912 – t 3 . Show that these polynomials fom1 a
bas is of IF’3.
23 . Le t B be the bas is of IF’3 con sisting of the Hermite po ly no-
mials in Exerci se 21, and let p (t) = 7 -12t – 8t 2 + 121 3 .
Find the coordinate vector of p relative to B.
24 . Let B be th e basis of IF’2 con sisti ng of the fir st three
Lagu erre polynomials listed in Exercise 22, and let
p (l) = 7 – 81 + 31 2 • F ind the coordinate vector of p rel ativ e
toB .
25 . Let S be a sub set of an n -dimensional vector space JI . and
sup pose S contains few er than n vect ors. Explain why S
ca nnot sp an V.
26 . Let H be an 11-climcnsional subspace of an n -dim ensional
vec to r sp ace V . Show that H = JI .
27 . Ex pla in why the space 1P’ of all pol ynomial s is an infinit e-
di me nsion al spac e.
2 See l 111 rod11 cti o 11 tu F11 11c ti o1w/ A11 a/_l’l’i s , 2nd ed .. by A . E. T;1 y lor ancl
David C . Lay (New Yo rk: John Wil ey & So ns, 198 0), pp . 92-93. Ot her
,e ls of po lynomi al~ arc di ~c u~sc cl th ere. 10 0 .
\

232 CHAPTER 4 Vector Sp acl~.s
28. Show that th-: spJc~ C UI~ ) of a~l cont~nuous functions defi ned
1..m the real Jin~ is an i111111ite-d11nens1onal space.
In Exncis..: s :29 .md 30, V is a nonzero finite-dimensional vector
“‘l ~,, ,111d the \’ ~cto rs listed belong to V . Mark each statement Sp. L~ , • .
True or Fulse. Ju stify each answer. (These questions are more
di f;ic ul t th:m those in Exercis es 19 and 20.)
29. a . If the re exists a set { V1, … , v p} that spans V, then
dim l’ ::: p.
b. If there exists a linearly independent set { v 1, . .. , v P} in
V , then dim V > p.
c . If dim V = p, then there exists a spanning set of p + I
vectors in V.
30. a. If there exists a linearly dependent set { v 1, .. . , v P} in V,
then dim V < p .
b. If every set of p elements in V fails to span V, then
dim V > p .
c . If p > 2 and dim V = p , then every set of p – I nonzero
vectors is linearly independent.
Exercises 31 and 32 concern finite-dimensional vector spaces V
and W and a linear transformation T : V -+ W .
31. Let H be a nonzero subspace of V, and let T (H) be the set of
images of vectors in H. Then T(H) is a subspace of_ W, by
Exercise 35 in Section 4 .2 . Prove that dim T(H) < dim H.
32. Let H be a nonzero subspace of V, and suppose T is
a one-to-one (linear) mapping of V into W . Prove that
dim T (H) = dim H. If T happens to be a one-to-o~e m~p-
ping of V onto W, then dim V = di m W . I~omo~pluc fimte-
dimensional vector spaces have the same d1mens10n.
33.
34.
11 a linearly independent set [M] According to Theorem a~ded to a basis for JR n. One } . JR ll can be e xp { v’ , … , v k m A – [ v I . . . v k e I . . . en ] ‘ · · t create –way to do this 1s O
f the identity matrix ; the pivot . the c olumns o w1the1, •• ·• e 11 . f IR?.11 f A form a basis or .
columns O
‘b d to extend the follow ing vectors Use the method descn e a . 5
to a basis for IR?. :
VJ=
– 9
-7
8
-5
7
9
4
6
-7
6
7
-8
5
– 7
Ex lain why the method works in general: ‘Y’hy are the
b. .P. . v included in the basis found for ongmal vect01 s v 1 , • •• , k
Col A ? Why is Col A = R 11 ?
[M] Let B = { 1 , cos t , cos2 t, … ‘ cos6 t} and C. = { 1, cos!’
cos 2t, . . . , cos 6t} . Assume the following tngonometnc
identities (see Exercise 37 in Section 4 .1) .
cos 2t = -1 + 2 cos2 t
cos 3t = -3 cost+ 4cos3 t
cos 4t = 1 – 8 cos2 t + 8 cos4 t
cos St = 5 cos t – 20 cos3 t + 16 cos5 t
cos 6t = -1 + 18 cos2 t – 48 cos4 t + 32 cos6 t
Let H be the subspace of functions spanned by the functions
in B. Then Bis a basis for H, by Exercise 38 in Section 4.3.
a. Write the B-coordinate vectors of the vectors in C, and
use them to show that C is a linearly independent set in H.
b. Explain why C is a basis for H .

3. vVhat is the next step to pcnu1~11 l,V …. ~ •- –helon form of AT? . , 1n a row ec 4. How many pivot columns dfC
4.6 E}{ERC~SES — 6 2 5 2 -3 In Exercises 1-4, assume that the matrix A is row equiv alent to B. -2 3 -3 -3 -4 Without calculations. list rank A and dim N ul A . Then fi nd bases 3. A == 9 5 9 ‘ 4 -6 for C ol A, Row A~ and Nul A. – 2 3 3 – 4 1
A=[-: -4 9
-7] 2 -3 6 2 5 1. 2 -4 1 ‘ 0 0 3 -1 1 5 -6 10 7 B == 0 0 0 1 3
B = [~
0
-~] 0 0 0 0 0 – 1
– 2 5
1 1 – 3 7 9 – 9 0 0
1 2 – 4 10 13 – 12 1 -3 4 – 1 9 4. A == 1 -1 -1 1 1 – 3 – 2 6 – 6 -1 – 10 1 – .3 1 – 5 -7 3 2 . A = 9 -6 – 6 – 3 ‘ 1 – 2 0 0 -5 -4 -3
3 – 9 4 9 0 1 1 -3 7 9 – 9 1 – 3 0 5 -7 0 1 – 1 3 4 -3 0 2 -3 8 B= 0 0 0 1 -1 – 2 0 B= 0 0 0 5 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0

– If a ~ x 8 matrix A has rank 3 fi nd er . N 1 :,. J .. k ;lr ‘ 1lll u ,1.d1 111RowA . an 1an .
(J If a 6 x 3 matrix A has ra nk J fi nd dii N 1 1 1 1′ k 1r · 11 u : .t1111Row 1, and r.in , .
7 Suppose a 4 x 7 matri x •t has r1111 r 1, i\’ ,t I I·. • “” “‘) . 1 · •\ 1:1 1 u11111 s . , Col A = .~ . ls Nul : t =- ll-R ‘! h .pln in )l’lll’ , 111:–,wrs .
S. SttPi’0″ 12 a :=; >.. (, matri\ I has t’l.l 1t r pivnt rnlu mns . W hat is di m Nul. I: ls Cnl. t = lk~ 1•.1 W hy nr wh y not ?
9, lf the null :–p,11..’ 1..’ tlf a ~ >.. (1 matrix A is 4-dimc nsi onal. wha t is the di1111..· 11s io11 (l f the co lumn space of ; I?
10. lf the nu ll sp ,h:c of a 7 x 6 matri x A is 5-dimensional, what is 111c dimensi on of the colu mn space of A?
11. lf th e null 11pace of an 8 x 5 matrix A is 2-di mensional , what is the dimension of the row space of A?
1~. If the null space of a 5 x 6 matrix A is 4-di mensional , what i, the dimension of the row space of A?
1.3. If .4 is a 7 x 5 matrix, what is the largest possible rank of A?
!f rl is a 5 x 7 matrix, wh at is the largest possible rank of A? Exp lain your answers.
l .t . If A is a 4 x 3 matrix, wh at is the largest possible dimension of the row space of A? If A is a 3 x 4 matrix, what is the largest possible dimension of the row sp ace of A? Explain.
15. If .4 is a 6 x 8 matri x , what is the smallest possible dimension of Nul A?
16. If A is a 6 x 4 matri x, what is the smallest possible dimension
of Nul A?
: 1 Exercise s 17 and 18, A is an m x n matri x . Mark each statement
True or False. Justify each an swer.
l 7. a. The row space of A is the same as the column space of
AT.
18.
b. If B is any echelo n form of A , an d if B has three nonzero
rows then the first three rows of A fonn a bas is for ‘
C.
d.
e.
a.
b.
C.
d.
Row A .
The dimensions of the row space and the column space
of A are the same , even if A is not square .
The <, um of the dimensions of the ro w space and the null
space of A equals the number of rows in A.
On a computer, row operations can change the app are nt
rank of a matrix.
If B is any echelon form of A, th en th e pivot columns of
B form a basis for the co lum n -, pace of A.
Row operations preserve the linear de pend ence relatio ns
among the ro ws of A. .
The dimension or the null !-. pace of ,1 is the number ol
col umns of A that arc not pivot colum ns.
The row space of Ar is the same as the column space of
A.
4.6 Rank 239
c. lf 1\ and B are row equ ivalent, the n their row spaces arc the ,amc .
19 . ~upposc the solutions of u homo gen eou s sys tem of five li ne ar c411atio n.., ll1 six un kno wns arc all mult iples of one no nze ro ,olutio11. Wi ll the sy -.t cm nece ssaril y have a sol ution for cwry pm,,ih lc choice o f co nsta nt s on the right side s of the equation .., ‘! Lxpl am.
20. Sup pose a nonhomogcncous sy~tc m of ‘>i x lin ear equ at io ns in ei ght unknow ns ha,; a ‘>o lut ion, with two fr ee variables. Is it possible to chan ge some constants on the equ ations ‘ right si des to make the new sys tem inco nsb,tc nt: Explain.
21. Suppose a no nho mo geneou s system of ni ne linear eq uations in ten unkn owns has a sol uti on for all possible cons tants on the righ t side s of the equ atio ns. Is it possible to fi nd two non zero so lutions of the ass ociated homogeneous system that are not multiples of each other: Discuss .
22. Is it possible that al l solut io ns of a ho mogeneo us system of ten linear eq uations in twe lve variables are multiples of one fixed non zero solution? Di scus s.
23. A homogeneous system of twelve li near eq uations in eight unknowns ha s two fixed so lutions that are no t multiples of each other, and all other sol utions are linear combinati ons of these two solutions. Can the set of al l solutions be de scribed with fewer than twelve homogeneous lin ear equations? If so. how many? Discuss.
24. Is it possible for a nonhomogeneous syste m of seven eq ua -tions in six unknowns to have a unique solution for some right-hand side of constants? Is it possible for such a system to have a unique solution for every right-hand side? Explain.
25 . A scientist solves a nonhomogeneous system of ten linear equations in twelve unknow ns an d fi nds that three of the unknowns are free variables. Can the scientist be certain that, if the right side s of the equatio ns are changed, the new nonhomogeneous system will have a sol ution? Disc uss .
26. In statistical theory, a common requirement is that a matrix be of full ra nk. That is, the rank should be as large as
possi ble. Explain why an m x n matrix with more rows than
columns has full rank if and only if its columns are linearly
independent.
Exercises 27-29 concern an m x n matrix A :md what are often
called the fun da menta l subspaces detennined by A .
27 .
28.
29.
T Which of the su bspaces Row .4. Co l A. Nu t A, Row A , Col AT. an d Nul AT are in !Rm an d whi ch are in lR. 11 ? How
many distinct subspaces arc in this list?
Justify th e fo ll o\\’ing equali ties:
a . Jim Row A + dim Nu! A = 11 ‘- tnnb , r o f L·o lunms or A
b. dim Col A+ d im Nu ! AT = m Numb er of rows of : \
Use faercise 28 to explain why the equation Ax = b has a so lution for all b in ~ 111 if and only if the equation AT x = 0
has only the trivial solution.

240 CHAPTER 4 \ cctor Sp.tees
30. Suppm,e , t ism ‘- 11 ~md bis in ~ 111 • What has to be tru e about
the t,\·o numb~rs rnnk l A b ] and rank A in order for the
cquJti(“‘tl Ax = b to be consistent?
R:.mk I matri~es are important in some computer algorithms and
se\ cral theoretical contexts , including the singular value decom-
l~osition in Chapter 7. It can be shown that an ,n x n matrix A
hJ~ r:nk 1 if and only if it is an outer product; that is, A = uvr
for some _u in lRm and v in ~ 11 • Exercises 31-33 suggest why this
property 1s true.
31. Verify th at rank uv7
< I if u = [ -n and v = [fl·
32. Letu = [; l Find v in R. 3 such that U =! : ] = uvr.
33. Let A be any 2 x 3 matrix such that rank A = 1, let u be the
first col umn of A, and suppose u # 0. Explain why there
is a vector v in JR 3 such that A = uvr. How could this
construction be modifi ed if the first column of A were zero?
34. Let A be an m x n matrix of rank r > 0 and let U be an ech-
elon form of A . Explain why there exists an invertible matrix
E such that A = EU , and use this factorization to write A
as the sum of r rank 1 matrices . [Hin t: See Theorem 10 in
Section 2.4 .]
7 -9 -4 5 3 -3 -7
-4 6 7 – 2 -6 -5 5
A 5 _ 7 -6 5 -6 2 8
35. [M] Let = 7 -3 5 8 -1 – -4 8
36.
37.
38.
6 -8 -5 4 4 9 3
Construct matrices C and N whose columns are bases for ,
a. Col A and Nul A, respectively, and construct a matrix R
whose rows form a basis for Row A.
b. Construct a matrix M whose column[sRtorm a ba.
sis for Nul Ar, form the matrices S = N ] and
T = [ c M ] , and explain why S and T should be
square. Verify that both S and T are invertible.
[M] Repeat Exercise 35 for a random integer-valued 6 x 7
matrix A whose rank is at most 4 . One way to make A
is to create a random integer-valued 6 x 4 matrix J and a
random integer-valued 4 x 7 matlix K, and set A = J K.
(See Supplementary Exercise 12 at the end of the chapter;
and see the Study Guide for matrix-generating programs.)
[M ] Let A be the matrix in Exercise 35. Construct a matrix
C whose columns are the pivot columns of A, and construct
a matrix R whose rows are the nonzero rows of the reduced
echelon fo rm of A . Compute CR, and discuss what you see.
[M] Repeat Exercise 37 for three random integer-valued
? x 7 matrices A whose ranks are 5, 4 , and 3. Make a con-
Jecture about how CR is related to A ~ . A p . 1or any matnx . rove your conJecture.

4 . 7 EX ERCISES
1. Let B = {b 1, b2 } and C = {c 1, c2 } be bases for a vector space
V , and suppose b 1 = 6c1 – 2c2 and b2 = 9c1 – 4c2.
a . Find the change-of-coordinates matrix from B to C.
b. Find [ x 1c for x = – 3b 1 + 2b2 . Use part (a).
2. Let B = {b 1, h2} and C = { c 1, c2 } be bases for a vector space
V, and suppose b 1 = -c1 + 4c2 and b2 = Sc 1 – 3c2 .
a. Find the change-of-coordinates matrix from B to C.
b. Find [ x 1c for x = Sb1 + 3b2 •

\ L,: t U == {u 1, U2} and W = {w 1 w } b b · · · · 2 e ases for V and let p be a matnx whose columns are [u ] [ ‘ . • 1 · . . 1 w nncl Uz ] w . Wluch of th e fol owmg equat10ns 1s satisfied by p f 11 . or a x 111 V?
(i) [x]u = P[xlw (ii) [ x)w = P [ x]u
4. Let A= {a 1, a2,a3 } and D = {d1.d2,d3} be bases fo r V,
and let P_ = [ _[di)~ _ [d2)..4 (d3]A ] . W hich of the fo ll ow-in g equat t0ns 1s sat1shcd by P for nll x in V’?
(i ) [x]A = P[x]v (ii) [ x )v = P [ x ]A
5. Let A={a 1, a 2.a., } and B ={b 1,b2, b 3 } be bases
for a ve ctor space V, and suppose a 1 = 4b 1 _ bz,
a2 = -b1 + b2 + b3.anda3 = b 2 -2b3.
a. Find the change-of-coordinates m atrix from A to B.
b. Find [x] 13 forx = 3a1 + 4a2 + a 3 •
6. Let V = {d 1. d2, d3} and F = {f1, f2 , f3 } be bases for
a vector space V, and suppose f 1 = 2d1 _dz + d3 ,
f2 = 3d 2 + d3, and f3 = – 3d 1 + 2d3 .
a. F in d the change-of-coordinates matrix from F to D.
b. Find [ x lv for x = f1 – 2f2 + 2f3 .
In Exercises 7-10, let B = {b1 , b 2} and C = {c 1, c2 } be bases for
3 2. In each exercise, fi nd the change-of-coordinates matrix from l3 to C and the change-of-coordinates matri x from C to B.
7. b 1 = [~].b2 = [ =n ,c1 = [ _;J. c2 = [-~ J
8. b 1 = [ – ~ l b2 = [ _; l C1 = [ ! l Cz = [ ~]
9. b 1 = [ =n ,b2 =[~l ei= [-ilc2 = [-~]
10. b1 = [ _;J.b2 = [-ilc1 = Ulc2 = [ ; ]
In Ex ercises 11 and 12, B and C are bases for a vec tor space V . \ti. ark each statement True or False. Justify each answer.
11. a. The columns of the change-of-coordinates matrix e!B are B-coordinate vectors of the vectors in C.
b. If V = ]Rn and C is the standard basis for V, then els is the same as the change-o f-coordinates matrix PB intro-
d uced in Section 4 .4.
12. a. The columns of els are linearly independent.
b. If V = JR 2, B = {b 1, b2}, and C = {c1,c2}, then row
reduction of [ c 1 c2 b 1 b2 ] to [ I P ] produces a
ma trix P that satisfies [ x ]13 = P [ x 1e for all x in V.
13. In 1P’2 , fin d the change-of-coordi nates matrix from the basis
B = {1 – 2t + t 2 , 3 – St+ 4t 2 , 2t + 3t 2} to the standard
bas is C = { 1, t . t 2 }. Then find the B-coordinate vector for
-] + 2t.
l 4, In lF’2 , fi nd the change-of-coordinates matrix from the ba-
sis B = { 1 – 3t 2 , 2 + t – 5t 2 , 1 + 2t} to the standard basis.
Then write t 2 as a linear combination of the polynomials in B.
4. 7 Change of Basis 245
Exerci ses 15 and 16 provide a proof of Theore m 15 . Fill in a
justifi cation for each step.
15 . Gi ven v in V, th ere exist scalars x1, ••• , x 11 , suc h that
because (a) __ . Apply the coordinate mapping deter-
min ed by th e basis C, and obtain
because (b) __ . This equation may be writte n in the form
(8)
by the definition of (c) _ _ . This sho ws that the matrix
elB shown in (5) satisfies [v]e = ef13 [v)13 for each v in V,
because the vector on the right side of (8) is (d) __ .
16. Suppose Q is any matrix such that
[v]c = Q[v)B for each v in V (9)
Set v = b I in (9) . Then (9) shows that (b 1] e is the first column of Q because (a) __ . Similarly, fork = 2, . . . , n, the kth col umn of Q is (b) __ because (c) _ _ . This shows
that the matrix clt3 defined by (5) in Theorem 15 is the only
matrix that satisfies condition (4).
17. [M] LetB = {xo, … ,x6}andC = {y0 , .. . ,y6},where xkis
the function cosk t and y k is the function cos kt. Exercise 34
in Section 4.5 showed that both B and C are bases for the
vector space H = Span {x 0 , .. . , x6}.
a . SetP =[[y0 ] 6 ( y GJ 6 ] ,andcalculateP-1.
b . Ex plain why the columns of p- 1 are the C-coordinate
vectors of Xo, . .. , x6 . Then use these coordinate vectors
to write trigonometric identities that express powers of
cost in terms of the functions in C.
See the Study Guide.
18. [M] (Calcu lus require d)3 Recall from calculus that inteorals b
such as
f (5cos3 t – 6cos4 t + 5cos5 t – 12cos6 t) dt (IO)
are tediou s to compute. (The usu al method is to apply inte-
gration by parts repeatedl y and use the half-angle formula .)
U se the matrix P or p – t from Exercis e 17 to transform (I O);
then compute the integral.
3 The idea for Exercises 17 and I 8 and five re lated exercises in earlier sections ca me from a p aper by Jack W. Rogers , Jr., of Auburn Uni versity, presented at a meet in g of th e Internation a l L inear Algebra Socie ty, Augu st 1995. See “Applications of Linear Algebra in Ca lculus.” American Mathematical Monthly 104 ( I), 1997.
1

246 CHAPTER 4 Vector Spaces
19. [~i] Lt>l
2
-5
6
-l] 0 )
J
a. Fi nd a basis {u 1• u2 , u3 } for Jffi. 3 such that P is the
cham!e-of-coordinates matrix from {u 1, u2, u3} to the
basis-{v 1• v2 . v3 } . [Hint: What do the columns of clB
:-epre sent ?]
20 .
b. Find a bas is {w 1, w2, w d for fl~ 1 wch tl”at p i ~ th(; cha n.;e –
of-coordin atcs matri x from { v 1, v2 , v ~} to f w. , ; r_; · w ~i}.
Let B = {b 1 b2 f1 C = fc c 1 ·111cl V – 1d d- I· b1..: be;.·.c.., ‘ ‘ I/• 2f, ‘ – l • ‘ –
for a two-d imens ional vector space .
W
. . , p p a. nte an equati on tha t rel ates th e ma tnCC ‘.) c – :3 • :;:::, -.: ·
and vf-B. Just ify you r res ul t.
b. [M] Use a matrix program eith er to help ~ ou f. nd rr- e
equ atio n or to check the equation you wTit c. \\”or~: \~ ith
three bases for JR 2 . (Se e Exe rci ses 7-1 0.)