Ebook: Linear Algebra Through Geometry

Linear Algebra Through Geometry introduces the concepts of linear algebra through the careful study of two and three-dimensional Euclidean geometry. This approach makes it possible to start with vectors, linear transformations, and matrices in the context of familiar plane geometry and to move directly to topics such as dot products, determinants, eigenvalues, and quadratic forms. The later chapters deal with n-dimensional Euclidean space and other finite-dimensional vector space. Topics include systems of linear equations in n variable, inner products, symmetric matrices, and quadratic forms. The final chapter treats application of linear algebra to differential systems, least square approximations and curvature of surfaces in three spaces. The only prerequisite for reading this book (with the exception of one section on systems of differential equations) are high school geometry, algebra, and introductory trigonometry.

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Linear Algebra Through Geometry introduces the concepts of linear algebra through the careful study of two and three-dimensional Euclidean geometry. This approach makes it possible to start with vectors, linear transformations, and matrices in the context of familiar plane geometry and to move directly to topics such as dot products, determinants, eigenvalues, and quadratic forms. The later chapters deal with n-dimensional Euclidean space and other finite-dimensional vector space. Topics include systems of linear equations in n variable, inner products, symmetric matrices, and quadratic forms. The final chapter treats application of linear algebra to differential systems, least square approximations and curvature of surfaces in three spaces. The only prerequisite for reading this book (with the exception of one section on systems of differential equations) are high school geometry, algebra, and introductory trigonometry.

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Linear Algebra Through Geometry introduces the concepts of linear algebra through the careful study of two and three-dimensional Euclidean geometry. This approach makes it possible to start with vectors, linear transformations, and matrices in the context of familiar plane geometry and to move directly to topics such as dot products, determinants, eigenvalues, and quadratic forms. The later chapters deal with n-dimensional Euclidean space and other finite-dimensional vector space. Topics include systems of linear equations in n variable, inner products, symmetric matrices, and quadratic forms. The final chapter treats application of linear algebra to differential systems, least square approximations and curvature of surfaces in three spaces. The only prerequisite for reading this book (with the exception of one section on systems of differential equations) are high school geometry, algebra, and introductory trigonometry.
Content:
Front Matter....Pages i-xii
Vectors in the Line....Pages 1-2
The Geometry of Vectors in the Plane....Pages 3-22
Transformations of the Plane....Pages 23-28
Linear Transformations and Matrices....Pages 29-38
Products of Linear Transformations....Pages 39-49
Inverses and Systems of Equations....Pages 50-60
Determinants....Pages 61-74
Eigenvalues....Pages 75-84
Classification of Conic Sections....Pages 85-97
Vector Geometry in 3-Space....Pages 98-112
Transformations of 3-Space....Pages 113-116
Linear Transformations and Matrices....Pages 117-121
Sums and Products of Linear Transformations....Pages 122-132
Inverses and Systems of Equations....Pages 133-150
Determinants....Pages 151-162
Eigenvalues....Pages 163-177
Symmetric Matrices....Pages 178-189
Classification of Quadric Surfaces....Pages 190-196
Vector Geometry in n-Space, n ? 4....Pages 197-204
Transformations of n-Space, n ? 4....Pages 205-212
Linear Transformations and Matrices....Pages 213-217
Homogeneous Systems of Equations in n-Space....Pages 218-225
Inhomogeneous Systems of Equations in n-Space....Pages 226-234
Vector Spaces....Pages 235-237
Bases and Dimension....Pages 238-244
Existence and Uniqueness of Solutions....Pages 245-246
The Matrix Relative to a Given Basis....Pages 247-252
Vector Spaces with an Inner Product....Pages 253-254
Orthonormal Bases....Pages 255-259
Orthogonal Decomposition of a Vector Space....Pages 260-262
Symmetric Matrices in n Dimensions....Pages 263-268
Quadratic Forms in n Variables....Pages 269-273
Differential Systems....Pages 274-290
Least Squares Approximation....Pages 291-295
Curvature of Function Graphs....Pages 296-301
Back Matter....Pages 303-307

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Linear Algebra Through Geometry introduces the concepts of linear algebra through the careful study of two and three-dimensional Euclidean geometry. This approach makes it possible to start with vectors, linear transformations, and matrices in the context of familiar plane geometry and to move directly to topics such as dot products, determinants, eigenvalues, and quadratic forms. The later chapters deal with n-dimensional Euclidean space and other finite-dimensional vector space. Topics include systems of linear equations in n variable, inner products, symmetric matrices, and quadratic forms. The final chapter treats application of linear algebra to differential systems, least square approximations and curvature of surfaces in three spaces. The only prerequisite for reading this book (with the exception of one section on systems of differential equations) are high school geometry, algebra, and introductory trigonometry.
Content:
Front Matter....Pages i-xii
Vectors in the Line....Pages 1-2
The Geometry of Vectors in the Plane....Pages 3-22
Transformations of the Plane....Pages 23-28
Linear Transformations and Matrices....Pages 29-38
Products of Linear Transformations....Pages 39-49
Inverses and Systems of Equations....Pages 50-60
Determinants....Pages 61-74
Eigenvalues....Pages 75-84
Classification of Conic Sections....Pages 85-97
Vector Geometry in 3-Space....Pages 98-112
Transformations of 3-Space....Pages 113-116
Linear Transformations and Matrices....Pages 117-121
Sums and Products of Linear Transformations....Pages 122-132
Inverses and Systems of Equations....Pages 133-150
Determinants....Pages 151-162
Eigenvalues....Pages 163-177
Symmetric Matrices....Pages 178-189
Classification of Quadric Surfaces....Pages 190-196
Vector Geometry in n-Space, n ? 4....Pages 197-204
Transformations of n-Space, n ? 4....Pages 205-212
Linear Transformations and Matrices....Pages 213-217
Homogeneous Systems of Equations in n-Space....Pages 218-225
Inhomogeneous Systems of Equations in n-Space....Pages 226-234
Vector Spaces....Pages 235-237
Bases and Dimension....Pages 238-244
Existence and Uniqueness of Solutions....Pages 245-246
The Matrix Relative to a Given Basis....Pages 247-252
Vector Spaces with an Inner Product....Pages 253-254
Orthonormal Bases....Pages 255-259
Orthogonal Decomposition of a Vector Space....Pages 260-262
Symmetric Matrices in n Dimensions....Pages 263-268
Quadratic Forms in n Variables....Pages 269-273
Differential Systems....Pages 274-290
Least Squares Approximation....Pages 291-295
Curvature of Function Graphs....Pages 296-301
Back Matter....Pages 303-307
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