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*Applied Linear Algebra*, by Noble and Daniels, 3rd Edition,
Prentice Hall (1988)

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Reviewed by Wiley Williams

Despite the name of this book, it is most appropriate for a second
course in Linear Algebra that emphasizes Matrix Analysis. The
overarching mathematical theme of the text is matrix
decomposition. The discussion often includes a discussion of how
modern software is used to solve matix problems.
Unusual features of the book include: Early chapters contain (in
addition to standard material) descriptions of block multiplication of
matrices, left and right inverses, motivational applications of linear
algebra, and an extensive discussion of gaussian elimination and the
LU-decomposition. After a treatment of vector spaces, independence,
basis and dimension, vector and matrix norms (over both the reals and
complexes) are discussed, the Banach Lemma is proved, and
ill-conditioning is discussed. Eigenvalues are motivated by the
vibrating string problem Three perspectives of eigenvalues and
eigenvectors are contrasted: diagonalizing a matrix, decomposing a
matrix in a form which makes computing powers easy, and finding a
basis so that the matrix of the associated linear transformation is
diagonal w.r.t the basis. A chapter on normal matrices and orthogonal
diagonalization is included. Orthogonal projections are discussed,
including Householder matrices, then the Gram-Schmidt process and its
implementation as the QR decomposition, as well as the use of this in
solving the least squares problem (which was one of the motivational
applications in Chapter 2). This culminates in a development of the
singular value decomposition and its use in solving the least squares
problem. A chapter on generalized eigenvalues and the Jordan Theorem.
A chapter on positive definite matrices, quadratic forms, and conic
sections. The text includes some problems to be solved using MATLAB,
but an instructor who wants to take full advantage of this technology
will need to provide a good deal of supplementation. Also, the problem
sets tend to not have enough routine problems or interesting
theoretical ones for students to fully explore the concepts involved.