A Guide to Advanced Linear Algebra
Linear algebra occupies a central place in modern mathematics. This book provides a rigorous and thorough development of linear algebra at an advanced level, and is directed at graduate students and professional mathematicians. It approaches linear algebra from an algebraic point of view, but its selection of topics is governed not only for their importance in linear algebra itself, but also for their applications throughout mathematics. Students in algebra, analysis, and topology will find much of interest and use to them, and the careful treatment and breadth of subject matter will make this book a valuable reference for mathematicians throughout their professional lives. Topics treated in this book include: vector spaces and linear transformations; dimension counting and applications; representation of linear transformations by matrices; duality; determinants and their uses; rational and especially Jordan canonical form; bilinear forms; inner product spaces; normal linear transformations and the spectral theorem; and an introduction to matrix groups as Lie groups. The book treats vector spaces in full generality, though it concentrates on the finite dimensional case. Also, it treats vector spaces over arbitrary fields, specializing to algebraically closed fields or to the fields of real and complex numbers as necessary.
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ˇ ˇ ˇ adjoint Advanced Linear Algebra bilinear form cA.x characteristic polynomial choose column complement conjugate Corollary cT.x defined Definition denote det.A det.xI diagonal entries diagonalizable dim.V dimensional direct sum divides easy to check eigenspace eigenvalue eigenvector element equivalent finite finite-dimensional vector space Gram-Schmidt process Guide to Advanced hence Hermitian Im.T induction inner product space integer invariant subspace invertible isometry isomorphism j j vn Jordan basis Jordan canonical form Ker.T Let B D Let T W V let W1 Lie group linear factors linear transformation linear transformation T W V linearly independent minimum polynomial monic polynomial mT.x n-by-n matrix nonsingular orthogonal pi.x polynomial p.x Proof quotient quotient spaces rational canonical form SLn.F space and let spaces and linear spans subset subspace ofV Suppose T-generated T-invariant unique V D W1 Þ Lemma Þ Remark