Linear Algebra and Matrix Theory

Real Coordinate Spaces: The Vector Spaces R[super]n. Subspaces of R[super]n. Geometric Interpretations of R[super]2 and R[super]3. Bases and Dimension. Elementary Operations on Vectors: Elementary Operations and Their Inverses. Elementary Operations and Linear Independence. Standard Bases for Subspaces. Matrix Multiplication: Matrices of Transition. Properties of Matrix Multiplication. Invertible Matrices. Column Operationsand Column-Echelon Forms. Row Operations and Row-Echelon Forms. Row and Column Equivalence. Rank and Equivalence. Vector Spaces, Matrices, and Linear Equations: Vector Spaces. Subspaces and Related Concepts. Isomorphisms of Vector Spaces. Standard Bases for Subspaces. Matrices over an Arbitrary Field. Systems of Linear Equations. Linear Transformations: Linear Transformations. Linear Transformations and Matrices. Change of Basis. Composition of Linear Transformations. Determinants: Permutations and Indices. The Definition of a Determinant. Cofactor Expansions. Elementary Operations and Cramer's Rule. Determinants and Matrix Multiplication. Eigenvalues and Eigenvectors: Eigenvalues and Eigenvectors. Eigenspaces and Similarity. Representation by a Diagonal Matrix. Functions of Vectors: Linear Functionals. Real Quadratic Forms. Orthogonal Matrices. Reduction of Real Quadratic Forms. Classification of Real Quadratic Forms. Bilinear Forms. Symmetric Bilinear Forms. Hermitian Forms. Inner Product Spaces: Inner Products. Norms and Distances. Orthonormal Bases. Orthogonal Complements. Isometries. Normal Matrices. Normal Linear Operators. Spectral Decompositions: Projections and Direct Sums. Spectral Decompositions. Minimal Polynomials and Spectral Decompositions. Nilpotent Transformations. The Jordan Canonical Form. Index.