Another specific of the book is that it is not written by or for an algebraist. So, I tried to emphasize the topics that are important for analysis, geometry, probability, etc. For example, I am only considering vector spaces over the fields of real or complex numbers.
Linear spaces over other fields are not considered at all, since I feel time required to introduce and explain abstract fields would be better spent on some more classical topics, which will be required in other disciplines. And later, when the students study general fields in an abstract algebra course they will understand that many of the constructions studied in this book will also work for general fields.
Also, I treat only finite-dimensional spaces in this book and a basis always means a finite basis. The reason is that it is impossible to say something non-trivial about infinite-dimensional spaces without introducing convergence, norms, completeness etc. And this is definitely a subject for a separate course text. So, I do not consider infinite Hamel bases here: they are not needed in most applications to analysis and geometry, and I feel they belong in an abstract algebra course.
In this version I also expanded a bit sections on non-orthogonal orthogonalization of the quadratic forms, and on singular value decomposition and its applications. In particular, I added a section about Moore--Penrose inverse Section 4.
Department home page. What is new: September version of the book includes correction of numerous typos. Some exercises were updated. In the September version I corrected numerous typos, noticed by the readers. I also added some more detailed explanations, in particular, clearly specifying in all situations whether real or complex case or both is considered. Text of the book September 4, PDF, 1. Below article will solve this puzzle of yours.
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