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chapter 1 linear equations in linear algebra 1introductory example: linear models in economics and engineering 11.1 systems of linear equations 21.2 row reduction and echelon forms 141.3 vector equations 281.4 the matrix equation ax = b 401.5 solution sets of linear systems 501.6 applications of linear systems 571.7 linear independence 651.8 introduction to linear transformations 731.9 the matrix of a linear transformations 821.10 linear models in business, science, and engineering 92supplementary exercises 102chapter 2 matrix algebra 105introductory example: computer models in aircraft design 1052.1 matrix operations 1072.2 the inverse of a matrix 1182.3 characterizations of invertible matrices 1282.4 partioned matrices 1342.5 matrix factorizations 1422.6 the leontief input-output modes 1522.7 applications to computer graphics 1582.8 subspaces of rn 1672.9 dimension and rank 176supplementary exercises 183chapter 3 determinants 185introductory example: determinants in analytic geometry 1853.1 introduction to determinants 1863.2 properties of determinants 1923.3 cramer¡¯s rule, volume, and linear transformations 201supplementary exercises 211chapter 4 vector spaces 215introductory example: space flight and control systems 2154.1 vector spaces and subspaces 2164.2 null space, column spaces, and linear transformations 2264.3 linearly independent sets: bases 2374.4 coordinate systems 2464.5 the dimension of a vector space 2564.6 rank 2624.7 change of basis 2714.8 applications to difference equations 2774.9 applications to markov chains 288supplementary exercises 299chapter 5 eigenvalues and eigenvectors 301introductory example: dynamical systems and spotted owls 3015.1 eigenvectors and eignevalues 3025.2 the characteristic equation 3105.3 diagonalization 3195.4 eigenvectors and linear transformations 3275.5 complex eigenvalues 3355.6 discrete dynamical systems 3425.7 applications to differential equations 3535.8 iterative estimates for eigenvalues 363supplementary exercises 370chapter 6 orthogonality and least squares 373introductory example: readjusting the north american datum 3736.1 inner product, length, and orthogonality 3756.2 orthogonal sets 3846.3 orthogonal projections 3946.4 the gram-schmidt process 4026.5 least-squares problems 4096.6 applications to linear models 4196.7 inner product spaces 4276.8 applications of inner product spaces 436supplementary exercises 444chapter 7 symmetric matrices and quadratic forms 447introductory example: multichannel image processing 4477.1 diagonalization of symmetric matices 4497.2 quadratic forms 4557.3 constrained optimization 4637.4 the singular value decomposition 4717.5 applications to image processing and statistics 482supplementary exercises 444appendixesa uniqueness of the reduced echelon form a1b complex numbers a3glossary a9answers to odd-numbered exercises a19index i1