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Linear Algebra

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Table of Contents

1 Linear Equations and Matrices.- 1.1 Systems of linear equations.- 1.2 Gaussian elimination.- 1.3 Sums and scalar multiplications of matrices.- 1.4 Products of matrices.- 1.5 Block matrices.- 1.6 Inverse matrices.- 1.7 Elementary matrices and finding A?1.- 1.8 LDU factorization.- 1.9 Applications.- 1.9.1 Cryptography.- 1.9.2 Electrical network.- 1.9.3 Leontief model.- 1.10 Exercises.- 2 Determinants.- 2.1 Basic properties of the determinant.- 2.2 Existence and uniqueness of the determinant.- 2.3 Cofactor expansion.- 2.4 Cramer's rule.- 2.5 Applications.- 2.5.1 Miscellaneous examples for determinants.- 2.5.2 Area and volume.- 2.6 Exercises.- 3 Vector Spaces.- 3.1 The n-space ?n and vector spaces.- 3.2 Subspaces.- 3.3 Bases.- 3.4 Dimensions.- 3.5 Row and column spaces.- 3.6 Rank and nullity.- 3.7 Bases for subspaces.- 3.8 Invertibility.- 3.9 Applications.- 3.9.1 Interpolation.- 3.9.2 The Wronskian.- 3.10 Exercises>.- 4 Linear Transformations.- 4.1 Basic propertiesof linear transformations.- 4.2 Invertiblelinear transformations.- 4.3 Matrices of linear transformations.- 4.4 Vector spaces of linear transformations.- 4.5 Change of bases.- 4.6 Similarity.- 4.7. Applications.- 4.7.1 Dual spaces and adjoint.- 4.7.2 Computer graphics.- 4.8 Exercises.- 5 Inner Product Spaces.- 5.1 Dot products and inner products.- 5.2 The lengths and angles of vectors.- 5.3 Matrix representations of inner products.- 5.4 Gram-Schmidt orthogonalization.- 5.5 Projections.- 5.6 Orthogonal projections.- 5.7 Relations of fundamental subspaces.- 5.8 Orthogonal matrices and isometries.- 5.9 Applications.- 5.9.1 Least squares solutions.- 5.9.2 Polynomial approximations 186.- 5.9.3 Orthogonalprojectionmatrices.- 5.10 Exercises.- 6 Diagonalization.- 6.1 Eigenvalues and eigenvectors.- 6.2 Diagonalization of matrices.- 6.3 Applications.- 6.3.1 Linear recurrence relations.- 6.3.2 Linear difference equations.- 6.3.3 Linear differential equations I.- 6.4 Exponential matrices.- 6.5 Applications continued.- 6.5.1 Linear differential equations II.- 6.6 Diagonalization of linear transformations.- 6.7 Exercises.- 7 Complex Vector Spaces.- 7.1 The n-space ?n and complex vector spaces.- 7.2 Hermitian and unitary matrices.- 7.3 Unitarily diagonalizable matrices.- 7.4 Normal matrices.- 7.5 Application.- 7.5.1 The spectral theorem.- 7.6 Exercises.- 8 Jordan Canonical Forms.- 8.1 Basic properties of Jordan canonical forms.- 8.2 Generalized eigenvectors.- 8.3 The power Ak and the exponential eA.- 8.4 Cayley-Hamilton theorem.- 8.5 The minimal polynomial of a matrix>.- 8.6 Applications.- 8.6.1 The power matrix Ak again.- 8.6.2 The exponential matrix eA again.- 8.6.3 Linear difference equations again.- 8.6.4 Linear differential equations again.- 8.7 Exercises.- 9 Quadratic Forms.- 9.1 Basic properties of quadratic forms.- 9.2 Diagonalization of quadratic forms.- 9.3 A classification of level surfaces.- 9.4 Characterizations of definite forms.- 9.5 Congruence relation.- 9.6 Bilinear and Hermitian forms.- 9.7 Diagonalization of bilinear or Hermitian forms.- 9.8 Applications.- 9.8.1 Extrema of real-valued functions on ?n.- 9.8.2 Constrained quadratic optimization.- 9.9 Exercises.- Selected Answers and Hints.

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From the reviews: "This is a standard book on Linear Algebra for science and engineering students. It covers the usual topics, including the Jordan canonical form, a topic that is omitted in many recent books at this level. The book reminded me of Strang's Linear Algebra and its Applications... Like Strang, the authors discuss linear difference and differential equations at some length, which should be useful to students in applied sciences. Unlike Strang, however, Kwak and Hong follow a more traditional line of presentation, with numbered definitions, lemmas, theorems, and examples. This may make it easier for the student to use the book as a reference. The exposition is clear but the style is not as chatty as Strang's. In summary, the book can be safely used as the basis for a course on Linear Algebra for the intended audience." --MAA Reviews "The emphasis is on computational skills along with mathematical abstractions; basic concepts are introduced by means of matrices and the solution of systems of linear equations. Many illustrative examples are given and all the usual advanced topics are treated ! The second edition has been substantially revised and new sections have been added." (ZENTRALBLATT MATH) "As linear algebra is one of the most important subjects in the study of science and engineering because of widespread applications in social or natural science, computer science, physics, or economics this book covers one of the most useful courses in undergraduate mathematics, providing essential tooks for industrial scientists... The primary purpose of the book is to give a careful presentation of the basic concepts of linear algebra as a coherent part of mathematics, and to illustrate its power and utility through applications of other disciplines. ---Educational Book Review

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