id: 06208463
dt: b
an: 2014b.00583
au: Farin, Gerald; Hansford, Dianne
ti: Practical linear algebra. A geometry toolbox. 3rd ed.
so: Boca Raton, FL: CRC Press (ISBN 978-1-4665-7956-9/hbk;
978-1-4665-7959-0/ebook). xvi, 498~p. (2014).
py: 2014
pu: Boca Raton, FL: CRC Press
la: EN
cc: H65 G75 N35 P25 R65
ut: 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 Bézier curve; computer graphics;
page ranking; image compression
ci: Zbl 1064.15001
li:
ab: 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 $\mathbb{R}$, well-illustrated with graphics and
hand-drawn 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 Bézier 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
undergraduate-level 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.
rv: John D. Dixon (Ottawa)