id: 06531029
dt: b
an: 2016e.00791
au: Lankham, Isaiah; Nachtergaele, Bruno; Schilling, Anne
ti: Linear algebra as an introduction to abstract mathematics.
so: Hackensack, NJ: World Scientific (ISBN 978-981-4730-35-8/hbk;
978-981-4723-77-0/pbk). x, 198~p. (2016).
py: 2016
pu: Hackensack, NJ: World Scientific
la: EN
cc: H65
ut: textbook; linear equation; vector space; linear transformation; fundamental
theorem of algebra; bases; null space; range; eigenvalue; eigenvector;
invariant subspace; determinant; norm; inner product space;
orthogonality; Gram-Schmidt orthogonalization; Hermitian operator;
unitary operator; normal operators; diagonalization; positive
operators; singular value decomposition; Gaussian elimination;
LU-factorization
ci: Zbl 1304.15001
li: doi:10.1142/9808
ab: The object of this book is “to introduce abstract mathematics and proofs
in the setting of linear algebra to students for whom this may be the
first step toward advanced mathematics”. Assuming some calculus and
only a very basic background in linear algebra (solution of linear
equations and manipulation of matrices), it provides a one-semester
course in finite-dimensional vector spaces and linear transformations
over the real and complex numbers up to the spectral decomposition
theorem for normal operators. The main part of the book consists of
eleven chapters, each roughly 10‒12 pages. The chapter headings are
as follows: What is linear algebra? Introduction to complex numbers
(operations, polar form and geometric interpretation); Fundamental
theorem of algebra (proof of the theorem, factoring polynomials over
$\mathbb{C}$); Vector spaces (spaces over $\mathbb{R}$ and
$\mathbb{C}$, subspaces and direct sums); Span and bases (linear
independence, dimension); Linear maps (null space and range, dimension
formula, matrices of linear maps with respect to bases); Eigenvalues
and eigenvectors (invariant subspaces, existence of eigenvalues over
$\mathbb{C}$, diagonal matrices, every complex linear operator has a
triangular matrix over a suitable basis); Permutations and determinants
(sign of permutation, definition of determinant, cofactor expansion);
Inner product spaces (norms, orthogonality, Gram-Schmidt
orthogonalization, orthogonal projections and minimization); Change of
bases (change of basis matrix for orthogonal bases); Spectral theorem
(Hermitian and unitary operators, normal operators, spectral theorem
and diagonalization, positive operators, singular value decomposition).
Each of the chapters is followed by “calculational exercises” to
test the reader’s understanding of the material and by
“proof-writing exercises” to develop the reader’s ability to
construct mathematical arguments of increasing difficulty. Perhaps
unusual in a book at this level is a proof of the fundamental theorem
of algebra based on the extreme value theorem for real-valued functions
of two real variables, and a proof of the existence of eigenvalues for
complex linear transformations without the use of determinants,
following {\it S. Axler} [Linear algebra done right. 3rd ed. Cham:
Springer (2015; Zbl 1304.15001)]. The use of Axler’s approach means
that an instructor could omit the chapter on permutations and
determinants without affecting the remainder of the book, and indeed,
the chapter on determinants is the least well written and may well have
been added as an afterthought. The reviewer has always been challenged
to know how to deal with the topic of determinants in a course such as
this. On one hand, determinants are ubiquitous in the mathematical
literature and the formula for a determinant in terms of permutations
is theoretically important, if not always essential, in many arguments.
On the other hand, this unintuitive formula is clumsy and does not lead
to conceptual proofs of fundamental properties of determinants such as
the multiplicative property. The last third of the book consists of
four appendices of which Appendix A, Supplementary notes on matrices
and linear systems, is by far the longest. Appendix A deals at length
with the relationship between linear transformations and matrices,
Gaussian elimination, elementary matrices, LU-factorization and
solution of linear equations. Appendices B and C summarize facts about
set theory and algebraic structures. Appendix D explains some history
of mathematical notation, use of logical symbols and who first
introduced the symbol $i=\sqrt{-1}$, for example. If you plan to teach
a course on linear algebra which emphasizes the theoretical side of the
subject, you might well consider this book.
rv: John D. Dixon (Ottawa)