id: 06049469
dt: b
an: 2014b.00582
au: Petersen, Peter
ti: Linear algebra.
so: Undergraduate Texts in Mathematics. Berlin: Springer (ISBN
978-1-4614-3611-9/hbk; 978-1-4614-3612-6/ebook). x, 433~p. (2012).
py: 2012
pu: Berlin: Springer
la: EN
cc: H65
ut: real vector space; complex vector space; inner product; spectral theorem;
linear operator; dual space; quotient space; minimal polynomial; Jordan
canoncial form; Frobenius canonical form; matrix exponential;
determinant; eigenvalue; eigenvector; linear differential equation;
matrix exponential; Gaussian elimination
ci:
li: doi:10.1007/978-1-4614-3612-6
ab: The five chapters of this text are: 1. Basic theory, 2. Linear operators,
3. Inner product spaces, 4. Linear operators on inner product spaces,
and 5. Determinants. The presentation is at the advanced undergraduate
level and includes the concepts needed by students who will continue on
to graduate studies. Among the topics included are the usual
introductory material on real and complex vector spaces, real and
complex inner products, the spectral theorem for normal operators, dual
spaces, quotient spaces, minimal polynomial, the Jordan canonical form,
the Frobenius canonical form, matrix exponentials, and determinants.
Given the clear and thorough discussion of inner products, it is a
missed opportunity not to have included material on the Fréchet-von
Neuman-Jordan theorem relating norms and inner products. The
presentation of material in the first four chapters is essentially
“determinant-free”. The principal tool to solve systems and
calculate eigenvalues is the Gaussian elimination. Throughout the text
applications to and connections with linear differential equations are
included. In the final chapter, the starting point for the concept of
the determinant of a linear operator $L:V \to V$ is the geometric
point-of-view that $\det \left( L \right)$ measures how $L$ changes
volumes in the vector space $V$. The discussions and examples are
clear, interesting, and appropriately thorough. There are numerous
well-chosen exercises to test the readers understanding and, in some
cases, to further develop some of the ideas. Although this is an
advanced text that is more suitable for a second linear algebra course,
it is nevertheless a text that should be included in every
undergraduate mathematics library. Even a beginning student will be
well-rewarded by exploring various topics in this book. [Minor
correction: on page 242 the vertical side of the figure should read $x
- {\text{pro}}{{\text{j}}_y}\left( x \right)$].
rv: F. J. Papp (Ann Arbor)