Practical Linear Algebra: A Geometry Toolbox, Third edition
Gerald Farin & Dianne Hansford
CRC Press, Taylor & Francis Group, An A K Peters Book
ISBN: 978-1-4665-7956-9


Errata for PLA 2nd edition

If you find errors in the text or figures, please report them to Dianne Hansford.
We will give you credit unless you request not to be named.

Chapter 1 Descartes' Discovery: 2D Lines Posted
page 12 Exercise 4: (min_1, min_2) should be (min_1, min_2, min_3)
and (max_1, max_2) should be (max_1, max_2, max_3)
Found by Jeff Griffin
page 13 Exercise 7: (v_1, v_2) should be (1/2,0). 02/03/17


Chapter 3 Lining Up: 2D Lines Posted
page 53 Line after (3.13): i = l(t-hat) should be i = l_1(t-hat) 09/06/2013


Chapter 5 2 x 2 Linear Systems Posted
page 105 In Example 5.5, the following sentence in misleading. "In this example, any vector u that is perpendicular to a_1 will be projected to the zero vector."
Actually, vectors perpendicular to the row vectors are mapped to the zero vector. In this example, the row vectors are identical to the column vectors.
page 114 Example 5.9: Typesetting error in the line starting with A = V'V^{-1}.
The matrix V' is incorrect in this line. It should be [v'_1 v'_2], thus the first column is [-1 -1]^T and the second column is [1 -1]^T
Found by Tim at Univ of Colorado at Colorado Springs


Chapter 7 Eigen Things Posted
page 144 Example 7.3: All references to r_1 should be r_2. This includes r_2,1 and r_1,1, which should be r_2,2 and r_1,2, respectively. 09/06/2013
page 152 In the displayed equation for the matrix R, the element r_1,2 should be +0.3. 05/15/2021


Chapter 8 3D Geometry Posted
page 175 Exercise 18: Let's be more specific. Let u_1 be the projection of w into the subspace defined by v 09/06/2013


Chapter 9 Linear Maps in 3D Posted
page 185

First equation on page: v' = Sv should be v' = S_1v.

On the next line, "An a_i,j element", is refering to an element of S_1



Chapter 10 Affine Maps in 3D Posted
page 214 First paragraph of page: "The intersection point x is" should refer to x' 09/06/2013
page 217 Bottom pf page: "transfomation" is missing an "r" 09/06/2013
page 219 The matrix following (10.10) would be better written with the last row as [n_1 n_2 n_3 0]. This would match the pipeline structure of sending one projection matrix for all vertices. Also, this matrix will act on homogeneous points and vectors, whereas the version in the text operates on points only.
Found by Christoph Fuenfzig


Chapter 13 Alternative System Solvers Posted
page 287 Equation (13.8): ||v| should be ||v|| 09/26/2013
page 299 Misspelling of involutory in Exercise 2 and WYSK.
This also occurs on pages 281, 284, and in the index.
Found by Radu Grosu


Chapter 14 General Linear Spaces Posted
page 306 In Example 14.5, the vector v_2 should be [5 0 6 1]^T 05/15/2021
page 306 In Example 14.6, the vector v_3 should be [3 4 -3]^T 05/15/2021
page 314 In the construction of Pv, each term should be divided by <u_j,u_j > for j=1,...,k. For example, the first term should be
<v,u_1 > u_1 / <u_1,u_1 >
page 315 Bottom of page: Equation for b_4 is missing ||.|| in the denominator; however, the solution is correct because the normalization was done.
||.|| = sqrt(1/2)
page 316 Section 14.5 example of polynomials of fixed degree n:
We consider all polynomials of degree < n to be of degree n.
Instead of "fixed degree n" we could have said "of degree less than or equal to n"
Found by Raud Grosu


Chapter 15 Eigen Things Revisited Posted
page 330 At the top of the page, in the conclusion to Example 15.7,
r = 1r_2 + 1r_3.
page 331 For the basic power method presented here, we assume that the matrix A has n distinct eigenvalues, thus in the second sentence it should read |lambda_1| > |lambda_2| 05/15/2021
page 341 Two occurances of r^(1) = [1 1 1] should be r^(1) = [1 1 1]^T 09/26/2013
page 335 Figure 15.3 should have an arrow from node 4 to node 3. 04/30/2015


Chapter 16 The Singular Value Decomposition Posted
page 347 Matrix V displayed near the top of the page is actually V^T
Found by T. Boult
page 347 The previous errata entry should be ignored. The actual error is in the display of V^T in the middle of the page. In error, V is given in the matrix product. 5/15/2021
page 353 "A note on U: ... eigenvectors A^TA" should state
"A note on U: ... eigenvectors AA^T"
Found by Radu Grosu
page 366 Exercise 17: The second column of the matrix X should be
-2, -1, 0, 1, 2, 1, 1
to match the given solution.
Found by Radu Grosu
page 366 Correction to previous correction!
Exercise 17: The second column of the matrix X should be
-2, -1, 0, 1, 2, 1, -1
to match the given solution.
Found by Javier Cano


Chapter 20 Curves Posted
page 413 There are 51 points generated by t=0, 1/50, ..., 1
Found by Seth Kingsley
page 425 Last paragraph, bottom of page: equation f_2 = f_1 \wedge f_3 should be f_2 = f_3 \wedge f_1. (See page 426 erratum for explanation.) 09/26/2013
page 426 Equation (20.18): should be f_2 = f_3 \wedge f_1.
Convention has it that f_2 has the same orientation with respect to the curve as the second derivative. This ideas is expressed in Sketch 20.9. Also, convention has it to create a right-handed local frame.
page 428 Exercise 10: attach the curve c(t) at d(0) = d_0. 09/26/2013


Appendix B Selected Exercise Solutions Posted
page 444 Exercise 4: (v_1, v_2, v_3) closing parenthesis is missing 09/06/2013
page 446 Exercise 28: solution should be
[3 2]^T - 0.5*[1 -1]^T = [5/2 5/2]^T
Found by Seth Kingsley
page 448 Exercise 27: The last equation should be
m(t) = [2 3/2]^T + t[-1 4]^T
Found by Temaran
page 453 Exercise 8: "is mapped to x=" should be "is mapped to x' =" 09/06/2013
page 455 Exercise 12: Incorrect article for "an hyperboloid" and "an paraboloid" 09/06/2013
page 462 Exercise 2: The question asked for an affine map; it is
p' = A(p - x_1) + y_1
In this exercise, Ax_1 = Ay_1, thus the affine map is really just a linear map (a reflection).
page 465 Exercise 10: The solution given is for v = [1/3 1/3 -2/3]^T and n = [0 0 1]^T. For v and n in the problem definition,
v' = [1/3 2/3 -2/3]^T + 2/3[-1 0 0 ]^T = [-1/3 2/3 -2/3]^T
Found by Temaran
page 482 Exercise 1: In the matrix, the lambda=1. Therefore, the circle is
x^Tx - r^2 = 0.
page 483 Exercise 4: Delete the last sentence of the solution. (The rotated ellipse was already given at the beginning of the solution.) 09/06/2013
page 484 Exercise 15: The given scaling is incorrect.
The circle with with center at the origin and radius 2 is mapped to the ellipse with x_1 extents +/- 1/sqrt(2) and x_2 extents +/- 1/2 by the scaling matrix with diagonal elements s_11 = 1/(2sqrt(2)) and s_22 = 1/4.
Found by Radu Grosu
page 483 Exercise 9: The definition of D is incorrect. The 4 should be a 2. 09/06/2013
page 485 Exercise 5: To complete the solution, we should add
"thus the monomial form is a_0 + ta_1 + t^2a_2 + t^3 a_3"
page 486 Exercise 8: f_2 and f_3 are incorrect.
f_3 = [0 0 1]^T and f_2 = [-0.99 0.14 0]^T
page 486 Exercise 10: The equation c_ 2 = c_0 + c[d_0 - d_1] should be
c_2 = d_0 + c[d_0 - d_1]