Linear Algebra and Its Applications

As you can see, the formatting is prohibitive.

I would recommend this book to those who have learned linear algebra long ago and plan to pick up some back.This book gives an extremely concise introduction to the core of linear algebra(so not quite the best for beginners), without any redundant sentences in trivial calculation. Starting with linear map and ending in spectral resolution, different parts are united. It also advances with the times, include topics like duality, convexity, which is interesting from engineering perspective.For those who only learn some elementary algebra in undergraduate, this book helps to make a big picture and to show some deep reason behind that.

Good book. Well worked out proofs.

Excellent book.

Great style from a great mind---but a bit too much material.

good

Note: The first edition totals 250 pages, ends with Chapter 17, and contains the first eight appendices of the second edition. Here we are offered a first rate exposition by a first class mathematician. An amazing amount of information is packed into this comparatively slim text. Much as a field of nuggets, each step taken points to the direction of another gem. Not easy if one is unaccustomed to the ways and whys of mathematical proofs. However, if allowing for a dose of mathematical maturity, the book does teach and inspire . If the topic of linear algebra seems but a smattering of disconnected entities, then this (non-elementary) textbook might just turn that viewpoint around.(1) First chapter, a rapid-fire snapshot of fundamentals: isomorphisms, linear dependence, equivalence relations.Read: "...this shows forming a quotient space amounts to throwing away information..." (page 6).(2) Linearity and duality: "chapter two might strike the reader--as it does the author-- as quintessential tautology." (page 12). Theorem six, here, quite lovely along with its enlightening proof.(3) Note: application to solutions of Laplace equation. Note: third chapter, exercise seven, presents a beautiful problem (page 24).(4) Matrices, next. Quick and to the point. Following which:(5) Determinants: excellent presentation of concepts and proofs (The Steps: volume,orientation, permutations, Cramer, trace, similarity).(6) Spectral Theory, Eigenvalues: "....analyzes linear maps of a space into itself by decomposing them into their basic constituents." The proof of Spectral Theorem nicely laid out. That preparation behind us, the next chapter, Euclidean Structure, is painless. Jumping ahead to...(7) calculus (ninth chapter): Theorem four, a delight, as "...the importance of the result lies in the connection it establishes between determinant and trace." (page 99). Now, jump ahead to page 129-- the derivation of the integral relations from theorem fifteen to the last formula (page 132) will prove to be an invaluable adjunct for physicists.(8) The same can be said for the eleventh chapter (an adjunct for physicists) as it deals with kinematics and dynamics. Along with the previous chapter, Matrix Inequalities, this one is a favorite. We read: "The determinant of a positive matrix does not exceed the product of its diagonal elements," then we delight in proof ! Physicists will delight in that exposition, as interplay of mathematics and physics is displayed to the utmost.(9) Read the pithy statements; these statements are the glue which holds the sections together linguistically.An example: "Convexity is a primitive notion, based on nothing but the bare bones of the structure of linear spaces over the reals...". Later we encounter normed linear spaces, followed by their linear mappings (chapters 14 and 15).We revel in the presentation of algorithms and limitations on accuracy of solutions: noting a fine section on steepest descent. An appendix details tensor products, beautiful.(10) Concluding: A Review in the American Mathematical Monthly sums it up... "....makes an effort to communicate some of the more unintuitive, but nonetheless important concepts, in Linear Algebra that illustrate the mystery of the dazzling connections between abstract mathematics and the real world." (Nov. 2001). Finally, these words from Professor Lax: "It is instructive to recall that in the 1940's Linear Algebra was dead as a subject for research, it was ready to be entombed in textbooks."Thankfully, we have texts such as this which propel one (and prepare one) for further research.Whether your interests are pure or applied, mathematics or physics, this exposition is as delightful as any to peruse.Challenging, but, not too much so. In other words, accessible.Highly Recommended !

After reading it, I became to realize that I know nothing about linear algebra. My intuition clearly tells that this is an excellent book, but the only problem is that I can not fully understand it. Sometimes the author diverges a little bit, maybe not just a little bit, the reader should always keep in mind that the main line is the first 8 chapters. If you find that you can not continue reading anymore, it is a good time to review the first 8 chapters again.

It was a very good bargain.

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