Elementary Linear Algebra

Vectors and matrices; Fundamental operations with vectors; The dot product; An introduction to proofs; Fundamental operations with matrices; Matrix multiplication; Systems of linear equations; Solving systems of linear equations; Equivalent systems and rank; Row space of a matrix; Inverses of matrices; Elementary matrices; Determinants; Introduction to determinants; Using row reduction to calculate determinants; Finite dimensional vector spaces; Subspaces; Span; Linear independence; Basis and dimension; Constructing special bases; Co-ordinatization; Linear transformations and orthogonality; Introduction to linear transformations; The matrix of a linear transformation; The dimension theorem; Isomorphism; Orthogonality and the Gram-Schmidt process; Orthogonal complements; Eigenvalues and Eigenvectors; Introduction to Eigenvalues; Diagonalization; Orthogonal diagonalization; Complex vector spaces; Products; Complex vector spaces; Inner product spaces; Applications; Graph theory; Ohm's law; Least-squares approximations; Markov chains; Hill substitution: an introduction to coding theory; Function spaces; Rotation of axes; Differential equations; Quadratic forms; Numerical methods for solving systems; LDU decomposition; The power method for finding Eigenvalues.