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Mod-04 Lec-16 The Rank-Nullity-Dimension Theorem. Isomorphisms Between Vector Spaces
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Classroom Contents
Linear Algebra
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- 1 Mod-01 Lec-01 Introduction to the Course Contents.
- 2 Mod-01 Lec-02 Linear Equations
- 3 Mod-01 Lec-03a Equivalent Systems of Linear Equations I: Inverses of Elementary Row-operations
- 4 Mod-01 Lec-03b Equivalent Systems of Linear Equations II: Homogeneous Equations, Examples
- 5 Mod-01 Lec-04 Row-reduced Echelon Matrices
- 6 Mod-01 Lec-05 Row-reduced Echelon Matrices and Non-homogeneous Equations
- 7 Mod-01 Lec-06 Elementary Matrices, Homogeneous Equations and Non-homogeneous Equations
- 8 Mod-01 Lec-07 Invertible matrices, Homogeneous Equations Non-homogeneous Equations
- 9 Mod-02 Lec-08 Vector spaces
- 10 Mod-02 Lec-09 Elementary Properties in Vector Spaces. Subspaces
- 11 Mod-02 Lec-10 Subspaces (continued), Spanning Sets, Linear Independence, Dependence
- 12 Mod-03 Lec-11 Basis for a vector space
- 13 Mod-03 Lec-12 Dimension of a vector space
- 14 Mod-03 Lec-13 Dimensions of Sums of Subspaces
- 15 Mod-04 Lec-14 Linear Transformations
- 16 Mod-04 Lec-15 The Null Space and the Range Space of a Linear Transformation
- 17 Mod-04 Lec-16 The Rank-Nullity-Dimension Theorem. Isomorphisms Between Vector Spaces
- 18 Mod-04 Lec-17 Isomorphic Vector Spaces, Equality of the Row-rank and the Column-rank I
- 19 Mod-04 Lec-18 Equality of the Row-rank and the Column-rank II
- 20 Mod-05 Lec19 The Matrix of a Linear Transformation
- 21 Mod-05 Lec-20 Matrix for the Composition and the Inverse. Similarity Transformation
- 22 Mod-06 Lec-21 Linear Functionals. The Dual Space. Dual Basis I
- 23 Mod-06 Lec-22 Dual Basis II. Subspace Annihilators I
- 24 Mod-06 Lec-23 Subspace Annihilators II
- 25 Mod-06 Lec-24 The Double Dual. The Double Annihilator
- 26 Mod-06 Lec-25 The Transpose of a Linear Transformation. Matrices of a Linear
- 27 Mod-07 Lec-26 Eigenvalues and Eigenvectors of Linear Operators
- 28 Mod-07 Lec-27 Diagonalization of Linear Operators. A Characterization
- 29 Mod-07 Lec-28 The Minimal Polynomial
- 30 Mod-07 Lec-29 The Cayley-Hamilton Theorem
- 31 Mod-08 Lec-30 Invariant Subspaces
- 32 Mod-08 Lec-31 Triangulability, Diagonalization in Terms of the Minimal Polynomial
- 33 Mod-08 Lec-32 Independent Subspaces and Projection Operators
- 34 Mod-09 Lec-33 Direct Sum Decompositions and Projection Operators I
- 35 Mod-09 Lec-34 Direct Sum Decomposition and Projection Operators II
- 36 Mod-10 Lec-35 The Primary Decomposition Theorem and Jordan Decomposition
- 37 Mod-10 Lec-36 Cyclic Subspaces and Annihilators
- 38 Mod-10 Lec-37 The Cyclic Decomposition Theorem I
- 39 Mod-10 Lec-38 The Cyclic Decomposition Theorem II. The Rational Form
- 40 Mod-11 Lec-39 Inner Product Spaces
- 41 Mod-11 Lec-40 Norms on Vector spaces. The Gram-Schmidt Procedure I
- 42 Mod-11 Lec-41 The Gram-Schmidt Procedure II. The QR Decomposition.
- 43 Mod-11 Lec-42 Bessel's Inequality, Parseval's Indentity, Best Approximation
- 44 Mod-12 Lec-43 Best Approximation: Least Squares Solutions
- 45 Mod-12 Lec-44 Orthogonal Complementary Subspaces, Orthogonal Projections
- 46 Mod-12 Lec-45 Projection Theorem. Linear Functionals
- 47 Mod-13 Lec-46 The Adjoint Operator
- 48 Mod-13 Lec-47 Properties of the Adjoint Operation. Inner Product Space Isomorphism
- 49 Mod-14 Lec-48 Unitary Operators
- 50 Mod-14 Lec-49 Unitary operators II. Self-Adjoint Operators I.
- 51 Mod-14 Lec-50 Self-Adjoint Operators II - Spectral Theorem
- 52 Mod-14 Lec-51 Normal Operators - Spectral Theorem
