CHAPTER 2
Section 2.1

• Page 116, footnote, in displayed equation and in line -3: Use small bold bullet not multiplication-dot. Should match usage in line -2.
• Page 128, Figure 2.12: Change 6 Volts to 16 Volts at top of Circuit Problem I
  Section 2.2
• Page 147, EXAMPLE 5, SOLUTION, Line 2: replace \frac12 by -\frac14 (twice) to read: $\bu=(1,-3,2)=-\frac14(-4,12,-8)=\frac14\bw$
  Section 2.3
• Page 166, Line 10: ... and write Domain($f$) to signify the
• Page 173, last displayed equation above EXAMPLE 6, both e-sub-j should be boldfaced.
• Page 186, Exercise 11, Line 3: ...-(x_1, x_2, x_3)$
• Page 189, General Exercise 2.3.61, Line 2: call a point $\bw$ .... (bold $\bw$ to match use in rest of line)
  Section 2.4
• Page 199, Definition, Vector Space, Property 5, end of Line 1: make 0 boldface zero.
• Page 206, Example 9: Is the set of these three matrices ...
• Page 221, General Exercise 31, in last two lines, should read: See also next exercise.
• Page 221, General Exercise 32: Add at end: ... are arbitrary constants, fixed at the outset.

CHAPTER 3
Section 3.1
• Page 233, Second displayed equation: Replace $(\bA\bB)_{ij}$ with $(\bA\bb_j)_i$
• Page 233, After third displayed equation: change "a a" to "a" to read: This formula provides a second way of ...
• Page 237, Figure 3.1: Shade parts (a) and (b) as well as improve figures.
• Page 245, Line -4: where the column vector $\bb$ is $m \times 1$ and ....
• Page 258, General Exercise 70, End of Line 2: Would $\IR^{n \times n}$
  Section 3.2
• Page 274, bottom displayed matrix E_3, interchange entries (3,1) and (3,2), last row to read: 0 -3/2 1
• Page 275, top displayed matrix E_3^{-1}, interchange entries (3,1) and (3,2), last row to read: 0 3/2 1
• Page 275, EXAMPLE 10, displayed matrix, entry (2,3): Change 10 to 1
• Page 277, Second displayed matrix, entry (1,1): Change 5/3 to -5/3

CHAPTER 4
Section 4.1
• Page 305, Example 4, Solution, Line 1: Add a step before the second equal sign: \left[\matrix{2&1&2\\ 0&5/2&3\\ -4&2&1}\right]=
• Page 306, Line -2: Insert (2) to read = (2)(2)(-1)(101)=404
• Page 313, Figure 4.2: Move to right to align y-axis with Figures 4.3 and 4.4 below.
• Page 313, Figure 4.4: Move to another page.
• Page 314, line -2: |Det[ Au, Av]|= ... = |Det[u, v]|
• Page 314, line -1: Insert: See Theorem 2, p. 332.~\qed
• Page 323, Multiple-Choice Exercise 4.1.12: For what value of $t$ is ...
  Section 4.2
• Page 334, EXAMPLE 8, SOLUTION, last row in last displayed matrix is 0 5 -2 and last line on page is =(-1)(-12+50)=-38
• Page 335, Line -2 and -1 above EXAMPLE 9: .... and is left as General Exercise 40.
• Page 337, Planes in $\IR^3$, Line 3: ... on a plane are given as ....

CHAPTER 5
Section 5.1
• Page 369, EXAMPLE 15, matrix entry (1,4): Change -4 to 4
• Page 405, EXAMPLE 19, displayed matrix A: Change entry (1,3) from 3 to 0 and change column 4 entries from 14 3 0 14 0 31 to 5 13 10 29 5 71
• Page 386, Add more Computer Exercises 5.1
  Section 5.2
  Section 5.3
• Page 437, EXAMPLE 5: Is there a linear transformation that maps these vectors $\bu_1=(2, 3)$ to $\bbv_1=(-1, 5)$ and $\bu_2=(4, 1)$ to $\bbv_2=(3, 0)$.
• Page 437, SOLUTION, Last Displayed Equation: $$ \bA = \left[\matrix{ 1 & -1\cr -\frac12 & 2 }\right] \bU = \left[\matrix{ 2 & 4\cr 3 & 1 }\right] \bV = \left[\matrix{-1 & 3\cr 5 & 0 }\right] $$

CHAPTER 6
Section 6.1
• Page 469, Line 1: However, for $\bB = ....$
• Page 469, Line 4: Finally, for $\bC = ....$
• Page 469, Line 6: consisting of eigenvectors of $\bC$ ...
• Page 477, Line 11: Suppose that $\bA$ is the simple matrix

CHAPTER 7
Section 7.1
• Page 516, Theorem 3, Proof, Line 1, and Theorem 4: Bad spacing.
• Page 517, Figure 7.3: Fix $p = \alpha y$ (Fix alpha symbol.)
• Page 527, Line -7: The space of all $n\times n$ real matrices ...
• Page 532, General Exercise 7.1.34b: Last term should be $\cdots + ||\by||^2$ to read $||\bx + \by||^2 = ||\bx||^2 + 2\langle\bx,\by\rangle + ||\by||^2$.
• Page 532, General Exercise 7.1.35, Should read: Establish that if $\langle\bx, \by\rangle$is real, then we have $||\bx + i \by||^2 = ||i \bx + \by||^2 = ||\bx||^2 + ||\by||^2$.
  Section 7.2
• Page 561, Example 9, Solution: Change \bu to \bv 21 times.
• Page 561, Example 9, Solution, Line-9: Suppose that $\ba$ and $\bp$ are vectors in $\IR^n$, and $\ba\neq \b0$.
• page 562, Example 10: Improve spacing.

CHAPTER 8
Section 8.1
• Page 585, second displayed equation: Omit \bU^{-1}\bA\bU=
• LA2: Page 591, Add comment about Google algorithm:
  Google's algorithm for ranking of Web pages uses a "popularity contest" style of search. Other iterative ranking methods predate Google's PageRank algorithm by nearly sixty years. In his 1941 paper, Nobel Prize winning economist Wassily Leontief discussed an iterative method for ranking industries. In 1965, sociologist Charles Hubbell published an iterative method for ranking people. Computer scientist Jon Kleingerg developed the Hypertext Induced Topic algorithm for optimizing Web information and it was referenced by Google's Sergey Brin and Larry Page. See Franceschet~[2010], in which he says that "PageRank stands on the shoulders of giants."
  "Google PageRank-Like Algorithm Dates Back to 1941" by Massimo Franceschet, PhysOrg.com, Feb. 19, 2010.
  Section 8.1
• Page 610, next to last displayed equation: Insert \bA \sim ... before left matrix
• LA2: Page 618, add footnote to singular value: \footnote\ddag{Gene Howard Golub (1932--2007) created, with William Kahan, an algorithm for computing the singular value decomposition. See Golub and Kahan~[1965]. The SVD algorithm is used in a wide variety of applicatons such as signal processing, data analysis, and Internet search engines. Because of the versiltatly of this algorithm, it has been called the {\sl Swiss Army Knife} of numerical computations and Gene was fondly known as {\sl Professor SVD}.}
• Page 620, COROLLARY 1, Line 1: Insert "and $\bA$ is a Hermitian matrix,"
• Page 620, COROLLARY 1, Line 2: Replace $\bA=\bU\bD\bU^{-1}$ with $\bA=\bU\bD\bU^H$
• Page 628, near bottom of page: Now we find that these vectors ...
  Section 8.3
• Page 651, Line -5 above EXAMPLE 3: L is the lower triangular part of ...
• LA2: Page 652: Add footnote: David M. Young, Jr., (1923-2008) established the mathematical foundation for the SOR method in his Ph.D. thesis at Harvard University. See Young~[1950]. Every since this ground-breaking research, iterative methods have been used on a wide range of scientific and engineering applications for solinvg large sparse systems of linear systems with new many new iterative methods (a.k.a. {\bf preconditioners}) having been developed. Young was one of the pioneers in modern numerical analysis and scientific computing. His car license plate read: {\sl Dr. SOR}.
• Page 660, Lemma 1: Bad spacing.
• Page 660, Lemma 2. Body of Lemma 2 should be in italics to match Lemma 1.

APPENDICES
Appendix A
• Page 687, A.5 Mathematical Induction, New wording: An important technique in establishing theorems is called {\bf mathematical inductions}. This is often used when a proposition is true for all positive integers. Let $P(n)$ be a proposition that depends on a positive integer, $n$. We want to establish $P(n)$ for $n=1,2,3,\ldots$.
• Page 688, Definition Strong Induction, end of Line 1: omit comma and at beginning of Line 2: replace first "then" with "\Rightarrow" to match one used in Definition Weak Induction
• Page 698, General Exercise 54: Add $\IZ$, $\IN$
• Page 698, General Exercise 56: $\sum_{k=1}^n a_k b_k = S_n b_n + \sum_{k=1}^{n-1} S_k (b_k - b_{k+1})$ where $S_i = a_1 + a_2 + \cdots + a_i$, $S_0 = 0$, and $1\leq n$.
  Appendix B

ANSWERS

• Page 706, Answer 1.1.33: Simplify to 15x - 12y = 13.
• Page 712, Answer 2.4.33: Omit "See Example 2 for definitions."
• Page 716, Answer 5.1. 39 c: [1 0 0 2], [0 0 1 6]
• Page 716, Answer 5.1. 39 f: Yes.
• Page 718, Answer 5.2.91: Omit.
• Page 721, Answer 7.2.43: Multiply by ...
• Page 722, Answer 8.1.53: ...is not invertible.

Acknowledgements: We welcome comments and suggestions concerning either the textbook or solution manuals. Send email to kincaid@cs.utexas.edu .