# Linear Algebra Video Lectures

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