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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: AA1 A1A 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 58 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 1417 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 1516. Should read, “In exercises 1516, 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 34 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 22n3 1 In rows 2 through 5: change 2 2 n1 to 1 4n 2 1 to 4 n1 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