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