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nakos g elementary linear algebra applications matlab maplesoft 2024

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Textbook in PDF format This text offers a unique balance of theory and a variety of standard and new applications along with solved technology-aided problems. The book includes the fundamental mathematical theory, as well as a wide range of applications, numerical methods, projects, and technology-assisted problems and solutions in Maple, Mathematica, and MatLAB. Some of the applications are new, some are unique, and some are discussed in an essay. There is a variety of exercises which include True/False questions, questions that require proofs, and questions that require computations. The goal is to provide the student with is a solid foundation of the mathematical theory and an appreciation of some of the important real-life applications. Emphasis is given on geometry, matrix transformations, orthogonality, and least-squares. Designed for maximum flexibility, it is written for a one-semester/two semester course at the sophomore or junior level for students of mathematics or science. Linear systems Introduction to linear systems Linear equations Definition of linear system Solution of linear system Geometry of solutions in two variables Back-substitution Introduction to Gauss elimination Geometry of solutions in three variables Linear systems with complex numbers Interchanges in terms eliminations and scalings Gauss elimination Matrices in echelon form The Gauss elimination algorithm Solution algorithm for linear systems The Gauss–Jordan elimination algorithm Existence and uniqueness of solutions Homogeneous linear systems Numerical considerations Applications: Economics, Chemistry, Physics, Engineering Economics: The demand function, market equilibria Chemistry: Chemical solutions, balancing of reactions Physics and engineering: Circuits, heat conduction Traffic flow Statics and weight balancing Numerical solutions of linear systems Computational efficiency of row reduction Iterative methods Jacobi iteration Gauss–Seidel iteration Convergence Comparison of elimination and Gauss–Seidel iteration Numerical considerations: Ill-conditioning and pivoting Miniprojects Psychology: Animal intelligence Counting operations in Gauss elimination Archimedes’ cattle problem Technology-aided problems and answers Selected solutions with Mathematica Selected solutions with MatLAB Selected solutions with Maple Vectors Matrices and vectors Addition and scalar multiplication Transpose, symmetric and Hermitian matrices Special square matrices, trace Geometric Interpretation of vectors Application of linear combinations Digital signals Matrix transformations The matrix–vector product Matrix transformations Matrix form of linear systems Relation between the solutions of Ax = 0 and Ax = b The span Linear independence Dot product, lines, hyperplanes Dot product Orthogonal projections Lines, planes, and hyperplanes Hyperplanes and solutions of linear systems Application: Computer graphics Plane matrix transformations Space matrix transformations Affine transformations Applications: Averaging, dynamical systems Data Smoothing by Averaging Discrete dynamical systems A population growth model Special topic: Tessellations in weather models Miniproject: Special affine transformations Technology-aided problems and answers Selected solutions with Mathematica Selected solutions with MatLAB Selected solutions with Maple Matrices Matrix multiplication Another viewpoint of matrix multiplication Powers of a square matrix Matrix multiplication with complex numbers Motivation for matrix multiplication Computational considerations Application to computer graphics Application to manufacturing Matrix inverse Computation of the inverse Relation of A−1 to square systems Properties of matrix inversion Matrix powers with negative exponents Application to statics: Stiffness of elastic beam Elementary matrices Elementary matrices and invertible matrices The matrix inversion algorithm Characterization of invertible matrices LU factorization Computational efficiency with LU LU with interchanges Block and sparse matrices Block matrices Addition of block matrices Multiplication of block matrices Inversion of block matrices Sparse matrices Applications: Leontief models, Markov chains Stochastic matrices Economics: Leontief input–output models Probability matrices and Markov processes Graph theory Graphs Sociology and psychology: Dominance graphs Miniprojects Cryptology: The Hill cipher Transition of probabilities Digraph walks A theoretical problem Technology-aided problems and answers Selected solutions with Mathematic Selected solutions with MatLAB Selected solutions with Maple Vector spaces Vector space Definition and properties Examples of vector spaces Subspaces Complex vector spaces Span, linear independence Span Linear dependence Linear independence Linear dependence for sequences Basis, dimension Basis of a vector space Dimension Ordered bases Coordinates, change of basis Coordinate vectors Change of basis Null space Column space, row space, rank The column space The row space Rank Rank and linear systems Application to coding theory Vector spaces over Z2 The Hamming (7, 4)-code Encoding and decoding Other types of codes Miniprojects Technology-aided problems and answers Selected solutions with Mathematica Selected solutions with MatLAB Selected solutions with Maple Linear transformations Linear transformations Evaluation of linear transformation from a basis Kernel and range Isomorphisms Matrix of linear transformation Change of basis and the matrix of a linear transformation Proof of dimension theorem The algebra of linear transformations Sums and scalar Products Composition of linear transformations Projections Linear transformation and matrix operations Invertible linear transformations Special topic: Fractals Fractals Miniproject Another fractal Technology-aided problems and answers Selected solutions with Mathematica Selected solutions with MatLAB Selected solutions with Maple Determinants Determinants: Basic concepts Cofactor expansion Geometric property of the determinant The Sarrus scheme for 3 × 3 determinants Properties of determinants Elementary operations and determinants Matrix operations and determinants The adjoint; Cramer’s rule Adjoint and inverse Cramer’s rule Determinants with permutations Permutations Computational consideration Applications: Geometry, polynomial systems Equations of geometric objects Elimination theory, resultants, and polynomial systems Special topic: Image recognition Introduction to projective geometry Projective transformations Projective invariants The object-image equations Miniprojects Vandermonde determinants Bezout resultant Technology-aided problems and answers Selected solutions with Mathematica Selected solutions with MatLAB Selected solutions with Maple Eigenvalues Eigenvalues and eigenvectors Computation of eigenvalues and eigenvectors Eigenvalues of linear operators Numerical note Diagonalization Diagonalization of matrices Powers of diagonalizable matrices An important change of variables Applications: Discrete dynamical systems Basic Concepts Long-term behavior of dynamical systems Uncoupling dynamical systems Applications: Dynamical systems (2) and Markov chains Dynamical systems with complex eigenvalues Application to a population growth problem Markov chains and stochastic matrices Limits of stochastic matrices Special topic: The PageRank algorithm of Google Approximations of eigenvalues The power method Rayleigh quotients (the Rayleigh–Ritz method) Origin shifts Inverse power method Shifted inverse power method Application to roots of polynomials Miniprojects The Cayley–Hamilton theorem Gerschgorin circles Transition of probabilities (Part 2) Technology-aided problems and answers Selected solutions with Mathematica Selected solutions with MatLAB Selected solutions with Maple Orthogonality and least squares Orthogonal sets and matrices Orthogonal sets Orthonormal sets Orthogonal matrices What makes orthogonal matrices important The Gram–Schmidt process Orthogonal complements Orthogonal projections and best approximation The Gram–Schmidt process Distance and angle from vector to subspace The QR factorization The QR method for eigenvalues Householder transformations and QR Least squares A least squares problem Solution of the least squares problem Least squares with QR factorization Inner product spaces Definition of inner product Examples of inner products Length and orthogonality Basic identities and inequalities The Gram–Schmidt process Complex inner products; unitary matrices Definition and examples Unitary matrices Polynomial and continuous least squares Polynomial least squares Continuous least squares (requires calculus) Special topic: The NFL rating of quarterbacks Miniprojects The Pauli spin and Dirac matrices Rigid motions in Rn Volume of the parallelepiped and the Gram determinant Technology-aided problems and answers Selected solutions with Mathematica Selected solutions with MatLAB Selected solutions with Maple Quadratic forms, SVD, wavelets Orthogonalization of symmetric matrices Proof of Schur’s decomposition theorem and example Quadratic Forms and conic sections Diagonalization of quadratic forms Applications of quadratic forms to geometry Positive and negative definite quadratic forms The singular value decomposition (SVD) Singular values; finding V, Σ, and U Pseudoinverse SVD and least squares The polar decomposition of a square matrix Special topic: SVD and image compression Fourier series and polynomials Application to wavelets Miniprojects Wavelets An image compression project generated by ChatGPT Technology-aided problems and answers Selected solutions with Mathematica Selected solutions with MatLAB Selected solutions with Maple Introduction to complex numbers Arithmetic with complex numbers Geometric interpretation of complex numbers Uniqueness of RREF Answers to selected exercises Bibliography Index of Applications Index

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nakos g elementary linear algebra applications matlab maplesoft 2024

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