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Systems of linear equations, Matrices, Elementary row operations, Row-reduced echelon matrices. Vector spaces, Subspaces, Bases and dimension, Ordered bases and coordinates.

Linear transformations, Rank-nullity theorem, Algebra of linear transformations, Isomorphism, Matrix representation, Linear functionals, Annihilator, Double dual, Transpose of a linear transformation.

Characteristic values and characteristic vectors of linear transformations, Diagonalizability, Minimal polynomial of a linear transformation, Cayley-Hamilton theorem, Invariant subspaces, Direct-sum decompositions, Invariant direct sums, The primary decomposition theorem, Cyclic subspaces and annihilators, Cyclic decomposition, Rational, Jordan forms.

Inner product spaces, Orthonormal bases, Gram-Schmidt process.

Lectures

Topic

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
  1. K.Hoffman and R. Kunze, Linear Algebra, 2nd Edition, Prentice- Hall of India, 2005.

  2. M. Artin, Algebra, Prentice-Hall of India, 2005.


  1. S. Axler, Linear Algebra Done Right, 2nd Edition, John-Wiley, 1999.

  2. S. Lang, Linear Algebra, Springer UTM, 1997.

  3. S. Kumaresan, Linear Algebra: A Geometric Approach, Prentice-Hall of India, 2004.



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