Advanced Linear Algebra by Bruce N. Cooperstein

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Advanced Linear Algebra by Bruce N. Cooperstein

PDF Free Download | Advanced Linear Algebra 2nd Edition by Bruce N. Cooperstein

Preface to Advanced Linear Algebra

The main differences between this edition and the first (apart from the correction of numerous typos) is the addition of a substantial amount of material, including four wholly new chapters.

As a consequence, through the choice of various subsets of the chapters, this book can be appropriate for a single upper-division or graduate course in linear algebra, or an upper-division or graduate sequence.

Furthermore, this book can function as a supplementary text for a graduate course on classical groups.

As with the first edition, the approach remains general (nearly everything is done over arbitrary fields) and structural.

We have also attempted to continue to build up to significant results from a few simple ideas. Following is a description of how the new edition specifically differs from its predecessor.

The first nine chapters of the edition have been carried over to the new edition with very few substantive changes.

The most obvious is renumbering: A chapter has been inserted between Chapters 8 and 9 so that Chapter 9 has now become Chapter 10.

Apart from the addition of several new exercises across these chapters, the most significant changes are:

Chapter 5 has been renamed “Normed and Inner Product Spaces” since we have added a section at the end of the chapter on “normed vector spaces”.

Here we introduce several norms that are not induced by an inner product such as the lp-norm for p ≥ 1 and the l∞-norm.

We show that all norms on a finite-dimensional real or complex space are equivalent, which implies that they induce the same topology.

In Chapter 8 we have added a section on orthogonal spaces over perfect fields of characteristic two and we prove Witt’s theorem for such spaces.

In Chapter 10 (previously 9), the fourth section on symmetric and exterior algebras has been split into two separate sections.

Additionally, we have added a section on Clifford algebras, which is a powerful tool for studying the structure of orthogonal spaces. The new chapters are as follows:

Chapter 8 is devoted to sesquilinear forms, which generalize the notion of a multilinear form.

In the first section we introduce the basic concepts, including the notion of a reflexive sesquilinear form and obtain a characterization: such forms are equivalent to Hermitian or skew-Hermitian forms.

In the second section we define what is meant by a unitary space, an isometry of a unitary space, and prove Witt’s theorem for non-degenerate unitary spaces.

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