Linear algebra

Description

Contents Preface 1 Systems of linear equations and matrices Systems of linear equations General systems of linear equations Matrices Row transformations and equivalence of matrices Row-echelon form Homogeneous systems 2 Matrix algebra Matrix arithmetic Inverse of a square matrix Properties of invertible matrices Matrix solutions of systems of linear equations Transpose of a matrix 3 Graphing calculators and matrices Matrix menu Inputting and editing a matrix Matrix arithmetic Calculating determinants Transpose of a matrix Solving linear systems using Gauss-Jordan elimination Solving linear systems using X = A[sup(–1)]C 4 Special types of square matrices Nonsingular matrices Triangular, diagonal, and scalar matrices Involutory, idempotent, and nilpotent matrices Symmetric and skew-symmetric matrices Orthogonal matrices Hermitian and skew-Hermitian matrices 5 Determinants Determinant of a square matrix Cramer’s rule Properties of determinant 6 Vectors in R[sup(n)] Vectors in two dimensions Dot product of vectors Vectors in R[sup(n)] Vectors as matrices 7 Vector spaces Definitions and terminology of vector spaces Linear independence Basis Dimension Row space, column space, and null space Rank and nullity 8 Inner product spaces Definition and terminology for inner product spaces Norm of a vector in an inner product space Cauchy-Schwarz inequality and properties of the norm Orthogonality in inner product spaces Gram-Schmidt procedure 9 Linear transformations Definition and terminology for linear transformations Kernel and image of a linear transformation Matrix representations of linear transformations Change of basis Algebra of linear transformations Linear operators on R2 and R3 10 Eigenvalues and eigenvectors The eigenvalue problem Useful properties of eigenvalues Diagonalization Answer key