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

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).

Chapter 2:

Here and There: Points and Vectors in 2D

Posted

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

Found by Ismail Keskin

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 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

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]

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

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

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.

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

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

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

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

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

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

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>

"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

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

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

Found by Aaron Arlet, Digipen Institute of Technology

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.

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

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