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.

1.0 Vectors in the Line.- 2.0 The Geometry of Vectors in the Plane.- 2.1 Transformations of the Plane.- 2.2 Linear Transformations and Matrices.- 2.3 Sums and Products of Linear Transformations.- 2.4 Inverses and Systems of Equations.- 2.5 Determinants.- 2.6 Eigenvalues.- 2.7 Classification of Conic Sections.- 3.0 Vector Geometry in 3-Space.- 3.1 Transformations of 3-Space.- 3.2 Linear Transformations and Matrices.- 3.3 Sums and Products of Linear Transformations.- 3.4 Inverses and Systems of Equations.- 3.5 Determinants.- 3.6 Eigenvalues.- 3.7 Symmetric Matrices.- 3.8 Classification of Quadric Surfaces.- 4.0 Vector Geometry in n-Space, n ? 4.- 4.1 Transformations of n-Space, n ? 4.- 4.2 Linear Transformations and Matrices.- 4.3 Homogeneous Systems of Equations in n-Space.- 4.4 Inhomogeneous Systems of Equations in n-Space.- 5.0 Vector Spaces.- 5.1 Bases and Dimensions.- 5.2 Existence and Uniqueness of Solutions.- 5.3 The Matrix Relative to a Given Basis.- 6.0 Vector Spaces with an Inner Product.- 6.1 Orthonormal Bases.- 6.2 Orthogonal Decomposition of a Vector Space.- 7.0 Symmetric Matrices in n Dimensions.- 7.1 Quadratic Forms in n Variables.- 8.0 Differential Systems.- 8.1 Least Squares Approximation.- 8.2 Curvature of Function Graphs.