Linear Algebra
Linear spaces, subspaces. Linear dependence, linear independence; span, basis, dimension, isomorphism. Quotient spaces. Linear functionals, Dual spaces. Linear mappings, null space, range, fundamental theorem of linear algebra. Underdetermined systems of linear equations. Composition, inverse, transpose of linear maps, algebra of linear maps. Similarity transformations. Matrices, matrix Multiplication, Matrix Inverse, Matrix Representation of Linear Maps, Determinant, Laplace expansion, Cramer's rule. Eigenvalue problem, eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem. Diagonalization. Spectral Theory for General Maps Finite Dimensions: The Eigenvalue Problem, Characteristic and Minimal Polynomials, Cayley-Hamilton Theorem, Spectral Mapping Theorem, Generalized Eigenvectors, Similarity Transformations, Similar Matrices. The Adjoint, Euclidean Structure on Linear Spaces. Vector norms, Orthogonal Projections & Complements, Orthonormal Basis, Matrix Norm, Isometry, Complex Euclidean Space. Spectral Theory for Selfadjoint Mappings, Quadratic Forms, Spectral Resolution, Orthogonal, Unitary, Symmetric, Hermitian, Skew-Symmetric, Skew-Hermitian and Positive Definite Matrices and Operators. Normal Maps, Commuting Maps and Simultaneous Diagonalization of Matrices. Rayleigh Quotient, The Minmax Principle.
