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Course11110605152 lectures

Linear Algebra

Dr. K.C. Sivakumar · IIT Madras

Start with lecture 1 At NPTEL

Lectures

SYSTEMS OF LINEAR EQUATIONS

  1. 1. Introduction to the Course Contents. Video
  2. 2. Linear Equations Video
  3. 3a. Equivalent Systems of Linear Equations I: Inverses of Elementary Row-operations, Row-equivalent matrices Video
  4. 3b. Equivalent Systems of Linear Equations II: Homogeneous Equations, Examples Video
  5. 4. Row-reduced Echelon Matrices Video
  6. 5. Row-reduced Echelon Matrices and Non-homogeneous Equations Video
  7. 6. Elementary Matrices, Homogeneous Equations and Non-homogeneous Equations Video
  8. 7. Invertible matrices, Homogeneous Equations Non-homogeneous Equations Video

VECTOR SPACES

  1. 8. Vector spaces Video
  2. 9. Elementary Properties in Vector Spaces. Subspaces Video
  3. 10. Subspaces (continued), Spanning Sets, Linear Independence, Dependence Video

BASIS AND DIMENSION

  1. 11. Basis for a vector space Video
  2. 12. Dimension of a vector space Video
  3. 13. Dimensions of Sums of Subspaces Video

LINEAR TRANSFORMATIONS

  1. 14. Linear Transformations Video
  2. 15. The Null Space and the Range Space of a Linear Transformation Video
  3. 16. The Rank-Nullity-Dimension Theorem. Isomorphisms Between Vector Spaces Video
  4. 17. Isomorphic Vector Spaces, Equality of the Row-rank and the Column-rank I Video
  5. 18. Equality of the Row-rank and the Column-rank II Video

MATRIX OF A LINEAR TRANSFORMATION

  1. 19. The Matrix of a Linear Transformation Video
  2. 20. Matrix for the Composition and the Inverse. Similarity Transformation Video

THE DUAL SPACE

  1. 21. Linear Functionals. The Dual Space. Dual Basis I Video
  2. 22. Dual Basis II. Subspace Annihilators I Video
  3. 23. Subspace Annihilators II Video
  4. 24. The Double Dual. The Double Annihilator Video
  5. 25. The Transpose of a Linear Transformation. Matrices of a Linear Transformation and its Transpose Video

EIGENVALUES AND EIGENVECTORS

  1. 26. Eigenvalues and Eigenvectors of Linear Operators Video
  2. 27. Diagonalization of Linear Operators. A Characterization Video
  3. 28. The Minimal Polynomial Video
  4. 29. The Cayley-Hamilton Theorem Video

INVARIANT SUBSPACES AND TRIANGULABILITY

  1. 30. Invariant Subspaces Video
  2. 31. Triangulability, Diagonalization in Terms of the Minimal Polynomial Video
  3. 32. Independent Subspaces and Projection Operators Video

DIRECT SUM DECOMPOSITIONS

  1. 33. Direct Sum Decompositions and Projection Operators I Video
  2. 34. Direct Sum Decomposition and Projection Operators II Video

PRIMARY AND CYCLIC DECOMPOSITION THEOREMS

  1. 35. The Primary Decomposition Theorem and Jordan Decomposition Video
  2. 36. Cyclic Subspaces and Annihilators Video
  3. 37. The Cyclic Decomposition Theorem I Video
  4. 38. The Cyclic Decomposition Theorem II. The Rational Form Video

INNER PRODUCT SPACES

  1. 39. Inner Product Spaces Video
  2. 40. Norms on Vector spaces. The Gram-Schmidt Procedure I Video
  3. 41. The Gram-Schmidt Procedure II. The QR Decomposition. Video
  4. 42. Bessel's Inequality, Parseval's Indentity, Best Approximation Video

BEST APPROXIMATION

  1. 43. Best Approximation: Least Squares Solutions Video
  2. 44. Orthogonal Complementary Subspaces, Orthogonal Projections Video
  3. 45. Projection Theorem. Linear Functionals Video

ADJOINT OF A LINEAR OPERATOR

  1. 46. The Adjoint Operator Video
  2. 47. Properties of the Adjoint Operation. Inner Product Space Isomorphism Video

SELF-ADJOINT, NORMAL AND UNITARY OPERATORS

  1. 48. Unitary Operators Video
  2. 49. Unitary operators II. Self-Adjoint Operators I. Video
  3. 50. Self-Adjoint Operators II - Spectral Theorem Video
  4. 51. Normal Operators - Spectral Theorem Video

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Madras
Department
Mathematics
Teacher
Dr. K.C. Sivakumar
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
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