1 Abelian Groups and Fields Definitions and examples of abelian groups and fields 2 Vector Spaces The definition of vector spaces, simple properties, and examples 3 Subspaces Subspaces of a vector space and linear combinations of vectors 4 Basis of a Vector Space Basis of a vector space, linear combination 5 Dimension of Vector Spaces Bases and dimension of vector spaces 6 Linear Maps The definition and examples of linear maps 7 Isomorphisms Equivalent vector spaces 8 Quotient Spaces Quotient spaces formed by modding out a subspace 9 Matrices Definition and operations of matrices 10 Space of Linear Maps Hom and Dual Space 11 Fundamental Theorem of Linear Algebra Fundamental theorem of linear algebra 12 Gaussian Elimination Gaussian elimination and inverse matrices 13 Determinant The definition and geometric meaning of the determinant 14 Existence and Uniqueness of the Determinant Existence and uniqueness proof of the determinant, and methods for computing it 15 Characteristic Polynomial The characteristic polynomial of a matrix 16 Eigenspace Decomposition Eigenspace decomposition of a vector space 17 Jordan Canonical Form Constructing Jordan form via generalized eigenspace decomposition 18 Minimal Polynomial Cayley-Hamilton theorem and minimal polynomial 19 Dual Space Dual spaces, dual maps, and orthogonal complements 20 Bilinear Forms Bilinear forms and dual spaces 21 Inner Product Spaces Properties of inner products over the real numbers 22 Complex Inner Product Spaces Hermitian inner products over complex numbers 23 Least Squares Method Orthogonal projections and least squares 24 Spectral Theorem Orthogonal diagonalization of self-adjoint operators 25 Complex Spectral Theorem unitary diagonalization of normal operators 26 Singular Value Decomposition Orthogonal decomposition of arbitrary real matrices 27 Quadratic Forms and Sylvester’s Law of Inertia Classification of real symmetric bilinear forms