Practical Linear Algebra
A Geometry Toolbox
, Second edition

by Gerald Farin & Dianne Hansford

published by A K Peters, Ltd., January 2005
384 pages, ISBN: 1-56881-234-5

Third edition available 20 August 2013

Errata

If you find errors in the text or figures, please email dianne@farinhansford.com!

Updated: 22 July 2013

Chapter 1: Descartes' Discovery Posted
page 10

Sketch 1.9 accompanying Example 1.3 is incorrect. Please see the Sketch below. (This was correct in the first edition -- The Geometry Toolbox.)

The correct Example to accompany the Sketch 1.9 in the text is as follows. Suppose that (u1, u2, u3) = (4/3, 3/2, 3/4). (That is our guess examining at the sketch.) Also, suppose that min = (0,1,0) and max = (1/2, 2. 1). Then (x1, x2, x2) = (2/3, 5/2, 3/4).

Found by Marco Frontini, Italy

9/9/05
 

The "corrected" solution given above for the sketch in the book seems to be incorrect!

(u1, u2, u3) = (0.75, 2.5, 0.5) and (x1, x2, x2) = (1.5, 1.5, 0.5).

3/6/13
page 14, Sketch 2.1 The point labeled as [-2, 2] should be [2,-2]. 7/22/13

Chapter 2:

Here and There: Points and Vectors in 2D

Posted

page 19

Equation (2.5) should be || kv || = |k| || v ||.

Found by Ismail Keskin

7/6/08
page 27 Second equation in page: the term k in the denominator should be |k|. This equation holds for nonzero k and w.

Found by Ismail Keskin
7/6/08

page 28

theta = acos(s)

s was defined in (2.14) as s = v . w, but here s should be s = v.w / (||v||||w||)

4/28/08

 

Chapter 4: Changing Shapes: Linear Maps in 2D Posted
page 61 Sketch 4.2: All labels v should be replaced by u

Found by Ismail Keskin
7/6/08
page 68
mid-page
Incorrect: e2'=[sin a, cos a]
Correct:   e2'=[-sin a, cos a]

Found by: Hiroshi Ashikaga, MD NIH/NHLBI
2/21/05
page 72 Exclude the zero vector for [a_1, a_2]^T in the definition of the
matrix [a_1, ca_2; a_2, ca_2]. As one linear map, this would not be a parallel projection.

Found by Marco Frontini, Italy
5/1/05
page 79 Figure 4.12: the reflection in the top and bottom sequence is about the e1 axis. The reference to the Figure, in Example 4.7 should state that the reflection is about the e1 axis.

Found by Ismail Keskin
7/6/08

 

Chapter 5:

2x2 Linear Systems

Posted

page 88

Just before equation (5.1), it reads "point vector equations", however it should read "vector equations"

3/6/13

page 100

The linear system that has first row elements 1 and 3 is incorrect. This linear system should be

[2, 1; 0, 0] u = [3; 0]

3/6/13
page 102 At mid-page, the equation |a_{1,1}| < |a_{1,2}| should be |a_{1,1}| < |a_{2,1}| 3/6/13

 

Chapter 6:

Moving Things Around: Affine Maps in 2D

Posted

page 115

In Example 6.4, the matrix A should be [-1, 0; 0, -1].

Found by Daniel Kurtz, Northeastern Univ., Boston

4/17/07

 

Chapter 7: Eigen Things Posted
page 125
after (7.2)
Replace "must be a projection" with "must be singular"

Found by: J. Dixon
1/30/06
page 131

The last sentence is wrong. A counterexample is given by the matrix [[0,1],[0,0]]. It has two zero eigenvalues but it is not the zero matrix. Its rank is one.

Found by: J. Dixon

1/30/06
page 134

Discussion of positive definite matrix should be expanded. (See PLA third edition)

If A is a symmetric matrix, then A_s = A.

A key concept is that these matrices lend themselves to numerically stable and efficient algorithms.

7/22/13
page 138, first paragraph

The singular values of A are the square roots of lambda_1 and lambda_2.

Found by: J. Dixon

Let the singular values be sigma_1 and sigma_2. The condition number of A is sigma_1/sigma_2.

1/30/06

 

7/21/13

Chapter 8: Breaking It Up: Triangles Posted
page 142
line 2
Incorrect: finite element analysis (FEM)
Correct: finite element method (FEM)

Found by: Hiroshi Ashikaga, MD NIH/NHLBI
3/6/05

 

Chapter 9 Conics Posted
page 162 At the bottom of the page, the expression for "c" is incorrect. It should be c = v^TAv - d 3/6/13
page 164,
exercise 5

The first term in the equation should be x_1 rather than x_2

Found by: A. Sicherer-Roetman, Maritime Research Institute Netherlands

8/11/08

 

Chapter 12: Linear Maps in 3D Posted
page 204
mid-page

Incorrect: left-hand-side of matrix equation reads [v_1, v_2, v_3]^T
Correct: [v_3, v_2, v_1]^T


Found by: Hiroshi Ashikaga, MD NIH/NHLBI

3/6/05
page 213, 2nd paragraph "triple Section" should be "Section" 7/22/13
page 213, 1st paragraph "A linear map A will change that volume to that of the skew box ..."
If the linear map is a rigid body motion, such as a rotation or uniform scale, then the a_i are orthogonal.
7/22/13

 

Chapter 13: Affine Maps in 3D Posted
page 230
line 3
Incorrect: n[q - x] = 0
Correct: n[q-x'] = 0

Found by: Hiroshi Ashikaga, MD NIH/NHLBI
3/6/05

 

Chapter 14: General Linear Systems Posted
page 250 The last linear system displayed on the page has "63" in the right hand side vector. This should be a "3". 3/6/13
page 254 Example 14.6: right-hand side should be [7, -1, 0]^T

Found by: Bill Petzke
10/29/06
page 260

The algorithm on this page can be tricky to follow. We have made an effort to write the book without summations, but in this case they might have made the algorithm easier to follow.

In the three equations, notice that one subscript for each u and l term of the equation runs from 1 up to k-1 or k-1 down to 1. When k=1, these elements do not exist. Look to the last term to determine how many terms will be in these equations.

For those familiar with summations, the first equations would be written
u_k,k = a_k,k - Sum (for m=1 to m=k-1) [l_k,m * u_m,k]

Found by: A. Sicherer-Roetman, Maritime Research Institute Netherlands

8/26/08
page 261

To make the example on this page clearer, and in light of the errata from the previous page, we could make it clear how two entries are computed.

u_1,1 = a_1,1 = 2
u_1,3 = a_1,3 = 4

Found by: A. Sicherer-Roetman, Maritime Research Institute Netherlands

8/26/08

 

Chapter 15: General Linear Spaces Posted
page 271

The last displayed equation should have minus signs rather than plus signs. The part that reads
<b_1,b_1> + ... + <u, b_r> should read <b_1,b_1> - ... - <u, b_r>

3/1/13
page 277 The roots to a degree 4 polynomial are too hard to find by hand. Ignore this exercise if you do not have software. (The solution is incorrect. See below for an explanation.) 7/21/13

 

Chapter 16: Numerical Methods Posted
page 290
footnote 1
Incorrect: "...that the sum..."
Correct: "...than the sum..."

Found by: Hiroshi Ashikaga, MD NIH/NHLBI
3/6/13
page 282 The sentence reading "For any n-vector ..." is a bit misleading. This statement is true for a_j where j=1,...,i. 3/6/13
page 283 The last line of the algorithm, the definition of \bar a_j is missing a transpose sign. 3/6/13
page 284 The sentence "Notice that a_3 ..." is incorrect. The vector a_3 is not changed because it is in the plane about which the reflection is occuring, thus since the Householder matrix has the involutary property, a_3 is not changed.  
page 292, Figure 16.3 The vectors should be labeled with boldface r_i 7/22/13

 

Chapter 17 Putting Lines Together: Polylines and Polygons Posted
page 302, Sketch 17.9 The top object is not a rhombus because it is not equilateral. 7/21/13
page 308

"a another way to check the area" -- omit "a"

Definition of n: u_2 listed twice, one should be u_3
(u_2 + u_3 + ... u_{n-2}). In denominator also.

Found by: A. Sicherer-Roetman, Maritime Research Institute Netherlands

8/29/08
page 309, example 17.3

Incorrect u_2 and u_3 and normal n
Correct: u_2 = [1, -1, 1]^T    u_3=[-1,1,1]^T    n=[0,0,1]^T

Found by: Daniel Kurtz, Northeastern Univ., Boston

5/17/07
     

 

Appendix B Selected Problem Solutions Posted
page 344, solution to problem 10 Solution for u^{perp} incorrect.
w - u = [3 ,2]^T - [1/2, -1/2]^T = [5/2, 5/2]^T

Found by Kyle Monroe, Digipen Institute of Technology
9/13/06
page 345, solution to problem 10

The line equation is incorrect. A correct line equation: l(t) = p + t (q - p)

Found by Aaron Arlet, Digipen Institute of Technology

9/21/06
page 345, solution to problem 11 The point r is incorrect, it should be r=[2,3/2]. (See problem 10 above.)

Thus m(t) = [2, 3/2]^T + t [-1, 4]^T

Found by Chris Tallman, Digipen Institute of Technology
9/21/06
page 349, solution to problem 3

The singular values of a matrix A are the square root of the the eigenvalues of A^TA.
The condition number of A is sigma_1/sigma_2, where sigma_i are the singular values of A.

The eigenvalues in the solution should be labeled lambda'_i, then sigma_1 = sqrt(lambda'_1) = 1.45 and sigma_2 = sqrt(lambda'_2) = 0.69.

The condition number of A is 1.45/0.69 = 2.10.

7/21/13
page 351,
solution to Chapter 9, problem 1
The matrix should be A = [1, -1; -1,1]. The rest of the solution is correct.

Found by: A. Sicherer-Roetman, Maritime Research Institute Netherlands
8/11/08
page 360,
solution to Chapter 14, problem 4

The solution given is incorrect; it should be x_2 = -0.202x_1 - 0.16.

We should have taken our own advice and sketched it! Here is a plot of the input points, the incorrect solution (thin line) and correct solution (thick line).

Found by: A. Sicherer-Roetman, Maritime Research Institute Netherlands

8/29/08
page 361, solution to Chapter 15, problem 11 Elementary row operations change the eigenvalues. A diagonalization method, such as SVD, can be used. (See PLA 3rd edition.) 7/21/13
page 364, solution to Chapter 17, problem 8

The correct normal is n = [0,0,1]^T. See Example 17.3 and its erratum above.

Found by: A. Sicherer-Roetman, Maritime Research Institute Netherlands

8/29/08

 

Index Instructor's Exercise Solutions Posted
Chapter 4, Exercise 4 The solution does not match the exercise question. 7/21/13

 

  Index Posted
Chapter 4, Exercise 4 Incorrect: symmetric matrix, 125, 282
Correct: symmetric matrix, 63, 125, 282
3/6/05