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A First Course in Linear Algebra
Published 2017Table of Contents: “…1 Systems of Equations -- 1.1 Systems of Equations, Geometry -- 1.2 Systems Of Equations, Algebraic Procedures -- 2 Matrices -- 2.1 Matrix Arithmetic -- 2.2 LU Factorization -- 3 Determinants -- 3.1 Basic Techniques and Properties -- 3.2 Applications of the Determinant -- 4 R^n -- 4.1 Vectors in R^n -- 4.2 Algebra in R^n -- 4.3 Geometric Meaning of Vector Addition -- 4.4 Length of a Vector -- 4.5 Geometric Meaning of Scalar Multiplication -- 4.6 Parametric Lines -- 4.7 The Dot Product -- 4.8 Planes in R^n -- 4.9 The Cross Product -- 4.10 Spanning, Linear Independence and Basis in R^n -- 4.11 Orthogonality and the Gram Schmidt Process -- 4.12 Applications -- 5 Linear Transformations -- 5.1 Linear Transformations -- 5.2 The Matrix of a Linear Transformation I -- 5.3 Properties of Linear Transformations -- 5.4 Special Linear Transformations in R^2 -- 5.5 One to One and Onto Transformations -- 5.6 Isomorphisms -- 5.7 The Kernel And Image Of A Linear Map -- 5.8 The Matrix of a Linear Transformation II -- 5.9 The General Solution of a Linear System -- 6 Complex Numbers -- 6.1 Complex Numbers -- 6.2 Polar Form -- 6.3 Roots of Complex Numbers -- 6.4 The Quadratic Formula -- 7 Spectral Theory -- 7.1 Eigenvalues and Eigenvectors of a Matrix -- 7.2 Diagonalization -- 7.3 Applications of Spectral Theory -- 7.4 Orthogonality -- 8 Some Curvilinear Coordinate Systems -- 8.1 Polar Coordinates and Polar Graphs -- 8.2 Spherical and Cylindrical Coordinates -- 9 Vector Spaces -- 9.1 Algebraic Considerations -- 9.2 Spanning Sets -- 9.3 Linear Independence -- 9.4 Subspaces and Basis -- 9.5 Sums and Intersections -- 9.6 Linear Transformations -- 9.7 Isomorphisms -- 9.8 The Kernel And Image Of A Linear Map -- 9.9 The Matrix of a Linear Transformation -- A Some Prerequisite Topics -- A.1 Sets and Set Notation -- A.2 Well Ordering and Induction -- B Selected Exercise Answers…”
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