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Linear Algebra Henry C. Pinkham Linear Algebra July 10, 2015 Springer Preface This is a textbook for a two-semester course on Linear Algebra. Although the pre- requisites for this book are a semester of multivariable calculus, in reality everything is developed from scratch and mathematical maturity is the real prerequisite. Tradi- tionally linear algebra is the first course in the math curriculum where students are asked to understand proofs, and this book emphasizes this point: it gives the back- ground to help students understand proofs and gives full proofs for all the theorems in the book. Why write a textbook for a two semester course? First semester textbooks tend to focus exclusively on matrices and matrix manipulation, while second semester textbooks tend to dismiss matrices as inferior tools. This segregation of matrix tech- niques on one hand, and linear transformations of the other tends to obscure the intimate relationship between the two. Students can enjoy the book without understanding all the proofs, as many nu- merically examples illustrate all the concepts. As is the case for most elementary textbooks on linear algebra, we only study finite dimensional vector spaces and restrict the scalars to real or complex numbers. We emphasize complex numbers and hermitian matrices, since the complex case is essential in understanding the real case. However, whenever possible, rather than writing one proof for the hermitian case that also works for the real symmetric case, they are treated in separate sections, so the student who is mainly interested in the real case, and knows little about complex numbers, can read on, skipping the sections devoted to the complex case. We spend more time that usual in studying systems of linear equations without using the matrix technology. This allows for flexibility that one loses when using matrices. We take advantage of this work to study families of linear inequalities, which is useful for the optional chapter on convexity and optimization at the end of the book. In the second chapter, we study matrices and Gaussian elimination in the usual way, while comparing with elimination in systems of equations from the first chap- ter. We also spend more time than usual on matrix multiplication: the rest of the book shows how essential it is to understanding linear algebra. v vi Then we study vector spaces and linear maps. We give the classical definition of the rank of a matrix: the largest size of a non-singular square submatrix, as well as the standard ones. We also prove other classic results on matrices that are often omitted in recent textbooks. We give a complete change of basis presentation in Chapter 5. In a portion of the book that can be omitted on first reading, we study duality and general bilinear forms. Then we study inner-product spaces: vector spaces with a positive definite scalar (or hermitian) product), in the usual way. We introduce the inner product late, because it is an additional piece of structure on a vector space. We replace it by duality in the early arguments where it can be used. Next we study linear operators on inner product space, a linear operator being a linear transformation from a vector space to itself, we study important special linear operators: symmetric, hermitian, orthogonal and unitary operatrps, dealing with the real and the complex operators separately Finally we define normal operators. Then with the goal of classifying linear operators we develop the important no- tion of polynomials of matrices. The elementary theory of polynomials in one vari- able, that most students will have already seen, is reviewed in an appendix. This leads us to the minimal polynomial of a linear operator, which allows us to establish the Jordan normal form in both the complex and real case. Only then do we turn to determinants. This book shows how much of the elemen- tary theory can be done without determinants, just using the rank and other similar tools. Our presentation of determinants is built on permutations, and our definition is the Leibnitz formula in terms of permutations. We then establish all the familiar theorems on determinants, but go a little further: we study the adjugate matrix and prove the classic Cauchy-Binet theorem. Next we study the characteristic polynomial of a linear operator, and prove the Cayley-Hamilton theorem. We establish the classic meaning of all the coefficients of the characteristic polynomial, not just the determinant and the trace. We conclude with the Spectral Theorem, the most important theorem of linear algebra. We have a few things to say about the importance of the computations of eigenvalues and eigenvectors. We derive all the classic tests for positive definite and positive semidefinite matrices. Next there is an optional chapter on polytopes, polyhedra and convexity, a natural outgrowth of our study of inequalities in the first chapter. This only involves real linear algebra. Finally, there is a chapter on the usefulness of linear algebra in the study of difference equations and linear ordinary differential equations. This only uses real linear algebra. There are three appendices. the first is the summary of the notation used in the boof; the second gives some mathematical background that occasionally proves use- ful, especially the review of complex numbers. The last appendix on polynomials is very important if you have not seen the material in it before. Extensive use of it is made in the study of the minimal polynomial. Leitfaden There are several pathways through the book. vii 1. Many readers with have seen the material of the first three sections of Chapter 1; Chapters 2, 3, 4 and 5 form the core of the book and should be read care- fully by everyone. I especially recommend a careful reading of the material on matrix multiplication in Chapter 2, since many of the arguments later on depend essentially on a good knowledge of it. 2. Chapter 6 on duality, and Chapter 7 on bilinear forms form an independent sec- tion that can be skipped in a one semester course. 3. Chapter 8 studies what we call inner-product spaces: either real vector spaces with a positive definite scalar product or complex vector spaces with a positive definite hermitian product. This begins our study of vector spaces equipped with a new bit of structure: an inner product. Chapter 9 studies operators on an in- ner product space. First it shows how to write all of them, and then it studies those that have a special structure with respect to the inner product. As already mentioned, the material for real vector spaces is presented independently for the reader who wants to focus on real vector spaces. These two chapter are essential. 4. In Chapter 9, we go back to the study of vector spaces without an inner prod- uct. The goal is to understand all operators, so in fact logically this could come before the material on operators on inner product spaces. After an introduction of the goals of the chapter, the theory of polynomials of matrices is developed. My goal is to convince the reader that there is nothing difficult here. The key result is the existence of the minimal polynomial of an operator. Then we can prove the primary decomposition and the Jordan canonical form, which allow us to decompose any linear operator into smaller building blocks that are easy to analyze. 5. Finally we approach the second main objective of linear algebra: the study of the eigenvalues and eigenvectors of a linear operator. This is done in three steps. First the determinant in Chapter 11, then the characteristic polynomial in Chapter 12, and finally the spectral theorem in Chapter 13. In the chapter concerning the spectral theorem we use the results on inner products and special operators of chapters 8 and 9 for the first time. It is essential to get to this material in a one semester course, which may require skipping items 2 and 4. Some applications show the importance of eigenvector computation. 6. Chapter 13 covers the method of least squares, one of the most important appli- cations of linear algebra. This is optional for a one-semester course. 7. Chapter 14, another optional chapter considers first an obvious generalization of linear algebra: affine geometry. This is useful in developing the theory of iinear inequalities. From there is an a small step to get to the beautiful theory of convex- ity, with an emphasis on the complex bodies that come from linear inequalities: polyhedra and polytopes. This is ideal for the second semester of a linear algebra course, or for a one-semester course that only studies real linear algebra. 8. Finally the material on systems of differential equations forms a good applica- tions for students who are familiar with multivariable calculus. 9. There are three appendices: first a catalog of the notation system used, then a brief review of some mathematics, including complex numbers, and what is most im- viii portant for us, the roots of polynomials with real or complex coefficients. Finally the last appendix carefully reviews polynomials in one variable. Recommended Books Like generations of writers of linear algebra textbooks before me, I must disclaim any originality in the establishment of the results of this book, most of which are at least a century old. Here is a list of texts that I have found very helpful in writing this book and that I recommend. • On the matrix side, I recommend three books: Gantmacher’s classic two volume text [8], very thorough and perhaps somewhat hard to read; Franklin’s concise and clear book [6]. Denis Serre’s beautiful book [24], very concise and elegant.
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