
06208463
b
2014b.00583
Farin, Gerald
Hansford, Dianne
Practical linear algebra. A geometry toolbox. 3rd ed.
Boca Raton, FL: CRC Press (ISBN 9781466579569/hbk; 9781466579590/ebook). xvi, 498~p. (2014).
2014
Boca Raton, FL: CRC Press
EN
H65
G75
N35
P25
R65
vector geometry
determinant
affine map
eigenvalue
eigenvector
algorithm
$LU$factorization
power method
nonnegative matrix
singular value decomposition
least squares
computational geometry
triangulation of a surface
fitting B\'ezier curve
computer graphics
page ranking
image compression
Zbl 1064.15001
The original edition of this book appeared in 2004 and was reviewed in [Zbl 1064.15001]. The current edition appears to be about 25\% longer than the first, but the general comments on the earlier edition remain true. The first half of the book describes vector geometry in two and three dimensions over $\mathbb{R}$, wellillustrated with graphics and handdrawn sketches. Determinants are introduced as signed areas or volumes, affine maps are defined and described geometrically, eigenvalues and eigenvectors computed for $2\times2$ matrices, and various forms of matrices such as rotations and shears are described. The next quarter of the book considers spaces of more general dimensions (still over $\mathbb{R}$) and describes algorithms such as $LU$factorization, iterative solution of systems of linear equations, the power method to find the largest eigenvalue of a nonnegative matrix, and the singular value decomposition (SVD) with applications to least squares. The last quarter of the book discusses the classification of conics and topics in computational geometry such as triangulation of a surface and fitting B\'{e}zier curves. In the second half of the book, there are some good descriptions of how linear algebra is applied. In the reviewer's opinion the more successful of these are: determining light and shading of surfaces in computer graphics, page ranking by Google using eigenvector computations, and image compression using SVD; each of these sections explains the application and gives a reasonable outline of what mathematics is involved. The authors claim: ``[By replacing] mathematical proofs with motivations, examples, or graphics \dots the book covers all of undergraduatelevel algebra in the classical sense." Unfortunately, the attempt at a wide coverage and the informal style means that some concepts introduced, particularly in the later parts of the book, are imprecise, and some assertions leave a suspicion that the authors may not be clear about the underlying mathematics themselves. For example, the statements that ``repeated eigenvalues reduce the number of eigenvectors" (p. 327), (nonparallel) eigenvectors of a (real) symmetric matrix are always orthogonal (p. 137), and that the eigenvalues of a (real) symmetric $2\times2$ matrix with positive determinant are both positive (p. 456) will confuse a thoughtful student who considers the matrix $I$. In the reviewer's opinion, this book would not be a good choice for mathematics majors, but might well be useful for a class (such as students in computer graphics) whose primary interest is an intuitive working knowledge of vector geometry in low dimensions.
John D. Dixon (Ottawa)