THE IDEA OF INVERSE Solving Systems of Linear Equations The Special Case of "Square" Systems GENERATING INVERTIBLE MATRICES A Brief Review of Gauss Elimination with Back Substitution Elementary Matrices The LU and LDU Factorization The Adjugate of a Matrix The Frame Algorithm and the Cayley-Hamilton Theorem SUBSPACES ASSOCIATED TO MATRICES Fundamental Subspaces A Deeper Look at Rank Direct Sums and Idempotents The Index of a Square Matrix Left and Right Inverses THE MOORE-PENROSE INVERSE Row Reduced Echelon Form and Matrix Equivalence The Hermite Echelon Form Full Rank Factorization The Moore-Penrose Inverse Solving Systems of Linear Equations Schur Complements Again GENERALIZED INVERSES The {1}-Inverse {1,2}-Inverses Constructing Other Generalized Inverses {2}-Inverses The Drazin Inverse The Group Inverse NORMS The Normed Linear Space Cn Matrix Norms INNER PRODUCTS The Inner Product Space Cn Orthogonal Sets of Vectors in Cn QR Factorization A Fundamental Theorem of Linear Algebra Minimum Norm Solutions Least Squares PROJECTIONS Orthogonal Projections The Geometry of Subspaces and the Algebra of Projections The Fundamental Projections of a Matrix Full Rank Factorizations of Projections Affine Projections Quotient Spaces SPECTRAL THEORY Eigenstuff The Spectral Theorem The Square Root and Polar Decomposition Theorems MATRIX DIAGONALIZATION Diagonalization with Respect to Equivalence Diagonalization with Respect to Similarity Diagonalization with Respect to a Unitary The Singular Value Decomposition JORDAN CANONICAL FORM Jordan Form and Generalized Eigenvectors The Smith Normal Form MULTILINEAR MATTERS Bilinear Forms Matrices Associated to Bilinear Forms Orthogonality Symmetric Bilinear Forms Congruence and Symmetric Matrices Skew-Symmetric Bilinear Forms Tensor Products of Matrices APPENDIX A: COMPLEX NUMBERS What is a Scalar? The System of Complex Numbers The Rules of Arithmetic in C Complex Conjugation, Modulus, and Distance The Polar Form of Complex Numbers Polynomials over C Postscript APPENDIX B: BASIC MATRIX OPERATIONS Introduction Matrix Addition Scalar Multiplication Matrix Multiplication Transpose Submatrices APPENDIX C: DETERMINANTS Motivation Defining Determinants Some Theorems about Determinants The Trace of a Square Matrix APPENDIX D: A REVIEW OF BASICS Spanning Linear Independence Basis and Dimension Change of Basis INDEX.