Reseña del libro "Linear Algebra For All Undergraduates (en Inglés)"
Blue Collar Scholar makes yet another wonderful out of print textbook available to everyone at a price everyone can afford with this outstanding linear algebra text. Murdoch's book was one of the first linear algebra textbooks that tried to give a broad presentation for well-prepared sophomores or juniors of average ability which was both mathematically careful and rich in applications. It's beautifully written, thorough and very well organized. This book emphasizes the relationship between linear algebra and the Euclidean and analytic geometry of the plane (and via isomorphism, the complex plane C) and three dimensional space.Murdoch completely avoids abstract vector spaces in the main text by limiting the scalar fields to R and C. It allows a careful presentation of the subject without recourse to abstract algebra concepts in a familiar context most students at this level should have at least a passing acquaintance with.Chapter 1 begins with vector spaces and their elementary properties, all strictly over the fieldsand C. No essential structure is lost in this manner that cannot be omitted in an introduction and the generalization to abstract fields is presented briefly in an appendix. Although the book's overall emphasis is on linear transformations, Chapters 2 and 3 give a virtually complete presentation of the theory and application of linear systems of n variables, n x n matrices and determinants. In a concrete presentation, this is the natural way to go because most applications and calculations in linear algebra are done via operations on matrices i.e. linear transformations with a choice of basis. However, this actual relationship between linear transformations and matrices is not explored explicitly until later, after a great deal of experience is obtained manipulating matrices. This furnishes a great many specific examples and provides experience with calculations. Chapters 4 and 5 develop classical geometry, inner products and orthogonality, relations (complements and projections) in real spaces .Chapter 6 finally defines and develops abstract linear transformations in low dimensions and the relationship to matrices under a change of basis. The critical theory of spectral analysis-i.e. the characteristic equation of a linear system and its solution producing eigenvalues as its roots and the resulting eigenvectors-is developed at length in chapter 7. It gives a detailed and very clear explication of the deduction of the Jordan form of a matrix-a critical topic many authors struggle to give an understandable but elementary discussion of. Chapter 8 gives applications of linear algebra to the analysis of quadratic forms, an old topic that usually isn't discussed much in basic courses anymore, but is quite important in both algebra and physics. The last chapter extends the previous results to complex vector spaces, where adjoints and Hermitian matrices come into play.In addition, Blue Collar Scholar founder and editor has added his usual personal 2018 edition preface, giving the historical background of this early textbook and the gradual emergence and evolution of the linear algebra course in American undergraduate mathematics programs. He has also added a substantial new Recommended Reading section, suggesting further reading in linear algebra for both teachers and students of this incredibly important subject.This careful, rich and inexpensive text is a fantastic choice for either a standard linear algebra text or self-study by undergraduates or graduate students in the physical or social sciences who need to learn linear algebra for either applications or the theory. The prerequisites are only high school algebra and geometry. A prior course in calculus, while helpful, is not really necessary. It should become a classic in rapid course and a popular selection for anyone and everyone interested in teaching or learning this beautiful and basic subject.
