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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

Systems of linear equations, equivalent systems, matrices

2

Elementary row operations, properties, examples

3

Row-reduced echelon matrices, elementary matrices,

4

Invertible matrices, elementary matrices, equivalence

5

Other equivalent conditions in terms of systems of linear equations, homogeneous, non-homogeneous, examples

6

Vector spaces, definition and examples

7

Subspaces, examples

8

Linear independence and dependence, examples

9

Basis, dimension, examples

10

Ordered basis, examples, coordinates

11

Linear transformations, examples

12

Range space, null space, examples

13

Injective, Surjective linear transformations, examples

14

Rank-nullity theorem, consequences, examples

15

Algebra of linear transformations

16

Isomorphism, examples

17

Matrix representation of linear transformations, examples

18

Change of bases, matrix representation, examples

19

Linear functionals, annihilator

20

Double dual, examples

21

Dual basis, properties, examples

22

Transpose of a linear transformation, properties

23

Characteristic Values and characteristic vectors, examples

24

Properties of characteristic values, vectors, examples

25

Similarity transformation, Diagonalizability, characterization

26

Minimal polynomial, properties and relationship with the characteristic polynomial

27

Cayley-Hamilton theorem, applications

28

Invariant subspaces, characteristic values, examples

29

Direct-sum decomposition, examples, properties

30

Invariant direct-sum decomposition, examples

31

The Primary decomposition Theorem I

32

The Primary decomposition Theorem II

33

Cyclic subspaces, annihilators, examples

34

Cyclic decomposition Theorem I

35

Cyclic decomposition Theorem II

36

Rational form, examples

37

Jordan form, examples

38

Inner product spaces, examples

39

Cauchy-Schwarz inequality, examples

40

Orthonormal bases, Gram-Schmidt procedure.
  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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