Introduction to the Course Contents – Linear Equations – Equivalent Systems of Linear Equations I: Inverses of Elementary Row-operations – Equivalent Systems of Linear Equations II: Homogeneous Equations, Examples – Row-reduced Echelon Matrices – Row-reduced Echelon Matrices and Non-homogeneous Equations – Elementary Matrices, Homogeneous Equations and Non-homogeneous Equations – Invertible matrices, Homogeneous Equations Non-homogeneous Equations – Elementary Properties in Vector Spaces. Subspaces – Subspaces (continued), Spanning Sets, Linear Independence, Dependence – Basis for a vector space – Dimension of a vector space – Dimensions of Sums of Subspaces – Linear Transformations – The Null Space and the Range Space of a Linear Transformation – The Rank-Nullity-Dimension Theorem. Isomorphisms Between Vector Spaces – Isomorphic Vector Spaces, Equality of the Row-rank and the Column-rank – The Matrix of a Linear Transformation – Matrix for the Composition and the Inverse. Similarity Transformation – Linear Functionals – The Dual Space – Dual Basis – Subspace Annihilators – Subspace Annihilators
The Transpose of a Linear Transformation. Matrices of a Linear – The Double Dual. The Double Annihilator – Eigenvalues and Eigenvectors of Linear Operators – Diagonalization of Linear Operators. A Characterization – The Minimal Polynomial – The Cayley-Hamilton Theorem – Invariant Subspaces – Triangulability, Diagonalization in Terms of the Minimal Polynomial – Independent Subspaces and Projection Operators – Direct Sum Decompositions and Projection Operators – The Primary Decomposition Theorem and Jordan Decomposition – Cyclic Subspaces and Annihilators – The Cyclic Decomposition Theorem – The Rational Form – Inner Product Spaces – Norms on Vector spaces – The Gram-Schmidt Procedure – The QR Decomposition – Bessel’s Inequality, Parseval’s Indentity, Best Approximation – Best Approximation: Least Squares Solutions – Orthogonal Complementary Subspaces, Orthogonal Projections – Projection Theorem. Linear Functionals – The Adjoint Operator – Properties of the Adjoint Operation. Inner Product Space Isomorphism – Unitary Operators – Self-Adjoint Operators – Spectral Theorem
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