Download Page 10, second line of right column

12/28/11
ERRATA FOR PRINTING 1 Of THE 10th Edition
Unless otherwise noted, the corrections that follow apply to the first printings of
both Elementary Linear Algebra, 10th Edition (ELA) and Elementary Linear Algebra
with Applications 10th Edition, (ELA+Apps). The printing number is the rightmost
digit at the bottom of the copyright page in the front matter.
Page x, Preface
In line 12 the number “2.2.8” should read “2.3.8”
Page xi, Preface of ELA, Reviewers and Contributors
“Przemslaw…” should read, “Przemyslaw…”
Page xii, Preface of ELA+Apps, Reviewers and Contributors
“Przemslaw…” should read, “Przemyslaw…”
Page 9, Exercises 7 and 8
Change the word “vector” to “3-tuple”
Page 10, Exercise 15, second line of right column
Delete “14” from end of line
Page 24, second line of exercise 33
Last inequality should be 0    2
Page 24, Exercise 40a
Change the start of the sentence to read, “If A is a matrix with three rows and five
columns, then, …”
Page 24, Exercise 40b
Change the start of the sentence to read, “If B is a matrix with three rows and six
columns, then, …”
Page 24, Exercise 40c
Change the start of the sentence to read, “If C is a matrix with five rows and three
columns, then, …”
Page 36, first line of Exercise 9
Change the sentence to read, “Referring to the matrices A and B in Exercise 7,”
Page 36, first line of Exercise 10
Change the sentence to read “Referring to the matrices A and B in Exercise 7,”
1
Page 41, first bulleted item
Change the sentence to read, “If ab = ac and …”
Page 43, last line of text above the Historical Note
Should read: AA1  A1A  I .
Page 46, first line of the proof of part © of Theorem 1.4.7
Should read: “Properties I and (m) of Theorem 1.4.1 imply that…”
Page 47, third line above the blue box
The sentence starting with “Part (e) is a less …” should read “Part (e) is less…”
Page 49, Exercises 20 and 21
Change the problem statement to read, “Repeat parts (a), (c), (d), (e), and (f) of Exercise
18 for the matrix”
Page 49, first line of exercise 26
The polynomial should be p( x)  x3  (a  b  c) x2  (ab  ae  be  cd ) x  a(be  cd )
Page 51, second set of numbered items
Number 3 should end with “…add c times r1 to r2.
Page 66, part (b) of the True/False
Change the first part of the sentence to read, “If A is a square matrix, and if the linear
system Ax = b has a unique solution, then…”
Page 69, second displayed formula
In the second grouping, change the first term to read, “aijbjj”
Page 70, delete the Margin Note entirely
Page 95, first text line below display (4)
Change “det[A]” to “det(A)”
Page 97, Title of Example 6
Change title to read, “Determinant of a Lower Triangular Matrix”, and near the end of the
first text line of Example 6, chance “upper triangular” to “lower triangular”
Page 98, second 3  3 matrix in Example 7
The entry in row 2 and column 1 should be 4
2
Page 99, direction line for Exercises 58
Reference should be to Equation (2).
Page 100, True-False
Delete question (g) and rename questions (h)(j) as (g)(i).
Page 102, first displayed formula
Following the first = sign in the display, change the expression “ka33C13” to read
“ka13C13”, and following the second = sign change the expression “a33C13” to read
“a13C13”
Page 105, Exercise 19
Change to read, “Repeat Exercises 1417 by using a combination of row operations and
cofactor expansion.
Page 110, Margin Note
Move the margin note to the bottom of p.111, and change the first two lines to read, “It
follows from Theorems 2.3.5 and 2.1.2 that if A is an invertible triangular matrix, then…”
Page 127, last line of Definition 4
Change the word “ ultiple” to read “multiple”.
Page 129, first line of Exercise 11
Change line to read, “Find an initial point P of a nonzero vector u = PQ with terminal
point Q(3, 0, –5) and such that”. [Put a vector bar on top of PQ].
Page 129, first line of Exercise 12
Change the line to read, “Find a terminal point Q of a nonzero vector u = PQ with initial
point P(–1, 3, –5) and such that”. [Put a vector bar on top of PQ].
Page 129, first part of Exercise 16
Change text to read, “Let v and w be the vectors in Exercise 15. Find…”
Page 138, Theorem 3.2.5 has extraneous hypothesis k
Change the theorem hypotheses to read: “If u, v, and w are vectors in Rn, then:”
Page 140, next to last sentence of Example 9.
Change the reference from Formula (8) to Formula (26).
Page 147, Figure 3.3.3
Change the label of the dot on the line L to read, “(cos
Page 149, third line of Proof (b)
Change the reference from Formula (16) to Formula (12)
3
Page 150 Concept Review
Add the bulleted item “Orthonormal set of vectors”
Page 153 last line before Theorem 3.4.2
Change the last sentence to read, “In summary, we have the following result.”
Page 156, first sentence following the dingbat that ends Example 4
Change the references from Equations (10) and (11) to Equations (9) and (10)
Page 159, direction line for Exercises 1516.
Should read, “In exercises 1516, find vector and parametric equations of the plane in R3
that passes through the origin and is orthogonal to v
Page 159, Exercise 15
Change the Hint to read, “Construct two nonparallel vectors orthogonal to v in R3 . ”
Page 178, Exercise 12
Change text to read, “The set of polynomials of the form a0  a1x with…”
Page 197, second line above Theorem 4.3.4
Change text to read, “…trix of (10) is zero for every such…”
Page 200, Exercises 24 and 25
In both exercises replace “… span a three dimensional subspace of F(– ,
“… are linearly independent in F(– ,
.”
.” with
Page 206, Example 8, Solution (b), second displayed formula
0 1
Change matrix that is multiplied by the factor “b”to read: 
0 0 
Page 213, last line of Proof ©
Change “dim(W) = dim(V)” to read: “W = V”
Page 223, Exercise 8(b)
Change to read, “…and use (12) to compute…”
Page 223, Exercise 10©
Change to read: “…and use (12) to compute…”
Page 223, Exercise 11(d)
Change to read: “…and use (12) to obtain…”
Page 257, Table 8
4
Change the parenthetical equation in the upper left white box to read, “ (0 £ k < 1) ”
Page 257, Table 8
Change the parenthetical equation in the lower left white box to read, “(k > 1)”
Page 259, Example 6, first displayed formula

cos2 ( / 6)
sin( / 6) cos(  / 6) 
First matrix after the “=” sign should read: 

sin 2 (  / 6)
sin( / 6) cos( / 6)

Page 259, end of the second line of the statement of Example 6
Change to read, “…with the positive x-axis.”
Page 260: First line of the statement of Example 7
Change the first part of the line to read, “Find the reflection of the vector x =(1, 5) about
the line …”
Page 260, Second line of the Solution of Example 7
Change to read, “standard matrix for the reflection is”
Page 262, Exercise 23
Change read, “Use Formula (15) to derive…”
Page 262, Exercise 27, fifth sentence
Change read, “Use Formula (15) to show that…”
Page 262, Exercise 28c
Change the problem to read, “Find all solutions of the equation in Exercise 27 that lie in
the interval [0, 2 , and then by substituting those solutions in Formula (15), find an
angle of rotation about the axis in part (b) that results from multiplication by the matrix A
in part (a).”
P270: First displayed equation on the page
Change reference number to “(10)”
P273, Exercise 29 (a)
Change to read, “What can you say about the range of the matrix…”
P274, line following display (1)
Change text to read, “This shows that multiplication by a diagonal 2 x 2 matrix with
nonnegative entries compresses or expands the…”
P275, 5th line of table (not counting the headings shaded blue)
5
k 0
Matrix should be 
 0 1 
P275, 6th line of table (not counting the headings shaded blue)
k 0
Matrix should be 
 0 1 
P275, 7th line of the table (not counting the headings shaded blue)
The label on the dot on the right part of the figure should be (1 + k, 1)
P275, 8th line of table (not counting the headings shaded blue)
Change the text in the leftmost box to read, “Shear in the y-direction with factor k > 0”,
and change the right part of the figure to be a parallelogram with vertices (0, 0), (0, 1),
(1, 1+ k), (1, k)
Page 280, lead in to Exercise 9
Change text to read, “Find the 2 x 2 matrix that shears by”
Page 280, lead in to Exercise 10
Change text to read, “Find the 2 x 2 matrix that compresses or expands by”
Page 281, lead in to Exercise 13 and to Exercise 14
Change the text to read, “In each part, find a single 2 x 2 matrix that performs…”
Page 284, first sentence of the last paragraph
The words, “state vector” in that sentence should be set boldface, italic.
Page 292, Supplementary Exercise #1
Change “…ordered pairs…” to read “…ordered triples”
Page 296, two lines above Theorem 5.1.1
Change the theorem reference to read, “Theorem 4.10.4”
Page 296, Figure 5.1.1
The second legend should read, “
”
Page 305, second line following Definition 1
Change “express B as B = Q–1AQ” to read “express A as A = Q–1BQ”
Page 307, last displayed formula at the end of Example 1
In second matrix, change the last entry in the first column from “21” to “1”
Page 311, second display in Example 6
Change the second equation to read,
rather than
6
Page 312, 3rd paragraph (just above Theorem 5.2.5), 4th line
Change to read: “…the characteristic polynomial of A is called the algebraic multiplicity
of 0.”
Page 329, Solution ©, first text line after the block of three displayed formulas
Fix to read: “so the solution satisfying these conditions is”
Page 329, Solution ©, second display
There is an “=” sign missing after y2
Page 329, Solution ©, third display
There is right bracket missing on the first matrix and the equal sign is too short
Page 330, in the box
The title of the box should be: A Procedure for Solving y  Ay if A is Diagonalizable
Page 332, Exercise 2(b)
The last equation should be y2 (0)  1.
Page 339, last line on the page
Should read:
<U, V> = tr(UTV) = u1v1 + u2v2 + u3v3 + u4v4
Page 344, Exercise 30, 3rd line
Should read: “on C[0, 1] to compute <f, g>.”
Page 349, line immediately above Theorem 6.2.5
Reference should read “4.8.8” rather than “6.2.5”
Page 351, Exercise #13,
Delete “w1 = (0, 0, 0, 0),” change “w2” to “w1” and change “w3” to “w2”; delete the
comma before the word “and” in the last sentence
Page 358, Theorem 6.3.5 Proof just above Step 1
Delete: “for EXSL”
Page 362, Theorem 6.3.8, second displayed formula
There should be primes on vs+1 and vk
Page 365, Exercise 18,
The third line should be changed to read, “…radians with the positive x-axis. Solve that”
Page 366, Exercise 26
7
Start exercise with the sentence: “Let R3 have the Euclidean inner product.”
Page 375, Exercise 15(d)
Last line should read: “…and check your result using Theorem 3.3.4.”
Page 396, Exercise 15(a)
Change the last two lines to read: “…is either a rotation or a reflection followed by a
rotation.”
Page 396, Exercise 15(b)
Change the second line to read: “and is a rotation or a reflection followed by a rotation if
det(A) = –1.”
Page 396, Exercise 16
Change the second and third lines of the statement to read: “…is a rotation or a reflection
followed by a rotation. Find the angle of rotation in either case.”
Page 403, Concept Review
In the first bulleted item of the second column, delete “(or eigenvalue decomposition)”
Page 404, Exercise 8
Delete P at beginning of exercise
Page 404, Exercise 19, last sentence
“you” should read “your”
Page 404, Exercise 20, formula for A
The coefficient “c” should read “c1”
Page 405, Theorem 7.2.4
Change the first line to read, “If A is an n ´ n matrix with real entries, then …”
Page 422, Proof of Theorem 7.4.1
First line should read, “…that xTAx has…”
P409: Replace the line immediately following Formula (11) with:
“If a, b, and c are not all zero, then the graphs of these equations in R3 are called central
quadrics and, more specifically, the graph of the second equation is called a central
quadric in standard position.”
Page 436, fourth line from the bottom of the page.
Change line to read: “We saw in Formula (12) of Section 4.9 that …”
Page 434, the marginal note should read:
“T(–v) = –T(v)”
8
Page 438, second displayed formula in Example 11
Change the display to read,
“D(f + g) = D(f) + D(g) and D(kf) = kD(f)”
Page 453, first two lines of Example 2.
Change the reference from Example 1 of Section 8.1 to Example 3 of Section 8.1.
Page 455, Solution of Example 4, line starting with “(verify).”:
Change “Formula (8)” to read: “Formula (7)”.
Page 457, Exercise 1(d).
First line should be: T1( x, y)  ( x  y, y, x),
Page 458, Exercise 22
Should read: “…be the linear transformations in Examples 11 and 12 of ….”
Page 460, Change the first line below Formula (5) to read
“We leave it as an exercise to show that in the special case where TC : Rn ® Rm is
multiplication by C, and …”
Page 460, Change Formula (6) to read
[TC ]B¢, B = C
Page 466, Exercise 6(c)
Final sentence should read: “If so, find the matrix of T 1 with respect to the basis B.”
Page 470, Change the beginning of Theorem 8.5.3 to read
“Two n ´ n matrices, A and B, are similar if and only if there exist two bases for Rn (one
for A and one for B) relative to which those matrices represent the same linear operator.
Moreover…”
Page 476, Exercise 22(b)
Last line should read: “…is in the kernel of L for all real values of c1 and c2 . ”
Page 486, Directions for Exercises 12-13
Change “an” to “the”
Page 495, instructions for Exercises 9-10
Change the last sentence to read: “Use Fomulas (9) and (10) to approximate the
dominant…”.
Page 501, instructions for Exercises 34
Delete the word “in” at the start of the second line.
9
Page 511, second displayed formula
0 
The matrix to the right of the equal sign should read:  
0 
Pages 556 and 557 of ELA+Apps, Tables 1 and 2
In the table heading with the diagonal line, the letter “x” should be set in boldface, roman
type (denoting a vector); the superscript is ok.
Page 565 of ELA+Apps, first matrix on the page
The 3rd letter in the blue label at the top should be “D” and not “F”
Page 654 of ELA+Apps
In the Prerequisites box change “Linear Transformations” to “Matrix Transformations”
Page 658 of ELA+Apps, Example 5
Reference should “Exercise 8”
Page 676 of ELA+Apps, Table 1
In the first two Age Intervals letter “ “ should be removed from the formulas
Where two page numbers are given below, the first applies to ELA and the second
to ELA+Apps.
Page 529/721, Exercise Set 1.2, #39
Change answer to read, “The nonhomogeneous system will have exactly one solution.”
Page 531/723, Exercise Set 1.4, #21
Delete the part letter (b) and the associated answer, but leave the part letters and answers
unchanged
Page 533/725, Exercise Set 1.6, #5
Should read: x = 1, y = 5, z = 1
Page 533/725, Exercise Set 1.6, True/False
Part (b) should read “False”
Page 535/727, Exercise Set 2.1, True/False
Delete answer (f) and re-letter subsequent solutions as (f)-(i)
Page 535/727, Exercise Set 2.3, true/false answers
(l) should read: “False”
Page 535/727, Chapter 2 Supplementary Exercise #13
10
Should read: b2  5b  21
Page 536/728, Exercise Set 3.1, #11(a)
Delete the “u =”, and change the answer to read, “P(–1, 2, –4) is one possible answer.”
Page 536/728, Exercise Set 3.1, #11(b)
Delete the “u =”, and change the answer to read, “P(7, –2, –6) is one possible answer.”
Page 537/729, Exercise Set 3.3, #7
Should read: “Yes”
Page 539/731, Exercise Set 4.3, #5(b)
Should read: “They do …” and not “The do”.
Page 539/731, Exercise Set 4.3, True/False (a)
Should read: “False”
Page 540/732, Exercise Set 4.5, True/False (i)
Should read: “True”
Page 546/738, True/False 5.1(a)
Should read: “False”
Page 546/738, Exercise Set 5.2, #21
1
0
First matrix should read: P  
0
0
0
1
0
0
0 0
1 1

1 0
0 1 
Page 546/738, Exercise Set 5.3, #5
Should read: “x = …” and not “x = …”
Page 546/738, Exercise Set 5.3, #23
 2 1
First matrix should read: P  
 2 0 
Page 546/738, Exercise Set 5.3, #25
 1 1
First matrix should be: P  
 1 0 
Page 549/741, Exercise Set 6.3, True/False
Part (b) should read: “False”
Page 549/741, Exercise Set 6.4, #7 (a), (b), (c)
11
Remove “hats” from all x’s.
Page 551/743, Exercise Set 7.2, True/False
Part (f) should read “True”
Page 553/745, Exercise Set 8.1 #9
Part after the semicolon should read: T (5,  3)  (35, 14)
Page 553/745, Exercise Set 8.1, #21
Part (b) should read: (4, 2, 6), (1, 1, 0), (3, 4, 9)
Page 553/745, Exercise Set 8.1, True/False (i)
Part (i) should read: “False”
Page 553/745, Exercise Set 8.2, #19
Should read: “Yes”
Page 555/747, Chapter 8 Supplementary Exercises, #19
Add part (c) answer: f ( x)  e x , g ( x)  e x
Page 556/748, Exercise Set 9.1, #11(a)
 2 0 0   1 12  12 

Should be: A  LU   2 1 0  0 0
1



1
 2 0 1 0 0
Page 556/748, Exercise Set 9.1, #11(b)
 1 0 0  2 0 0  1 12  12 

Should be: A  L1DU1   1 1 0  0 1 0  0 0
1




1
 1 0 1  0 0 1 0 0
Page 556/748, Exercise Set 9.1, #11(c)
 1 0 0   2 1 1
Should be: A  L2U 2   1 1 0   0 0 1



 1 0 1  0 0 1
Page 556/748, Exercise Set 9.2, #7(a)
 1 
First matrix should read: 
 0.5
Page 750, Exercise Set 10.2, #6(c)
Should read: “...for
> 8 is nonbinding and for...”
12
Page 759, Exercise Set 10.16, #6
In first matrix:
In first and last row: change
1
22n3
1
In rows 2 through 5: change
2
2 n1
to
1
4n 2
1
to
4
n1
Page 567/771 Index
Below the index entry for “Symmetric Matrices” change the page number for the
subentry “orthogonally diagonalizing” from “299” to “399”.
The following are corrections for the Student Solutions Manual
Page 19, Exercise Set 1.4, Exercise 21
Delete the solution to part (b), but leave the part labels and answers to (a), (c), (d), and (f)
unchanged
The following are corrections for the Instructor Solutions Manual
Page 20, Exercise Set 1.4, Exercise 20
Delete the solution to part (b), but leave the part labels and answers to (a), (c), (d), and (f)
unchanged
13