Department of MathematicsIndian Institute of Technology, Bombay

Powai, Mumbai–400076, INDIA.

MA 106 – Linear Algebra

Spring 2014

Instructors

Balmohan V. Limaye

Murali Srinivasan

Syllabus and Course Outline of MA 106

For the syllabus and the complete list of references see http://www.math.iitb.ac.in/academic/btech.pdf .

Reference

T. Apostol, “Calculus, Vol. II, 2nd Edition, John Wiley and Sons (1969).

E. Kreyszig, “Advanced Engineering Mathematics”, 8th Edition, John Wiley and Sons (1999).

Lecture Notes will be put up on Moodle at the end of each week.

Course Outline

1. Generalities about matrices, matrix multiplication, special kinds of matrices.

2. Systems of equations and Gauss elimination, Elementary Matrices, Inverse of a matrix.

3. Determinants (as multilinear, alternating, normalized functions), product rule.

4. Vector spaces, notions of linear independence and dependence, the linear span of a set of vectors,vector subspaces, basis of a vector subspace, the row space, null space, column space of a matrix,rank of a matrix.

5. Linear transformations, the rank-nullity theorem and the matrix of a linear transformation.

6. Inner products, the Gram-Schmidt process, orthonormal bases, projections and the least squaresapproximation.

7. Eigenvalues and eigenvectors, characteristic polynomials and the eigenvalues of special matrices (sym-metric, skew-symmetric, hermitian, orthogonal, unitary, normal), Algebraic and geometric multiplic-ities. Diagonalisation by similarity transformations and the spectral theorem.

Time TableClasses & TutorialsBatch Lecture Slots Lecture Timings Venue Tut. Slots Tut. Timings Tut. Venue

D1 2A Mon : 9:30-10:25

2B Tue : 10:35-11:30 Hall 1 3C Thu: 8:30-9:25 LCT 31,32,33,22,23

2C Thu : 11:35-12:30

D2 2A Mon : 9:30-10:25

2B Tue : 10:35-11:30 Hall 2 3C Thu: 8:30-9:25 LCT 34,35,36,25,26

2C Thu : 11:35-12:30

D3 9A Mon : 15:30-16:55

9B Thu : 15:30-16:55 Hall 1 X2 Wed: 15:00-15:55 LCT 31,32,33,22,23

D4 9A Mon : 15:30-16:55

9B Thu : 15:30-16:55 Hall 2 X2 Wed: 15:00-15:55 LCT 34,35,36,25,26

Office Hours/ Extra ClassesSlot 7A (Wednesday : 8:30 A.M.- 9:25 A.M. ) for D1 & D3Slot 7B (Friday : 8:30 A.M.- 9:25 A.M. ) for D2 & D4

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Time Table for Tutorials

Venue Batch Timings

LCT 31 D1 T1 Thu : 8:30-9:25

LCT 32 D1 T2 Thu : 8:30-9:25

LCT 33 D1 T3 Thu : 8:30-9:25

LCT 22 D1 T4 Thu : 8:30-9:25

LCT 23 D1 T5 Thu : 8:30-9:25

LCT 34 D2 T1 Thu : 8:30-9:25

LCT 35 D2 T2 Thu : 8:30-9:25

LCT 36 D2 T3 Thu : 8:30-9:25

LCT 25 D2 T4 Thu : 8:30-9:25

LCT 26 D2 T5 Thu : 8:30-9:25

LCT 31 D3 T1 Wed: 15:00-15:55

LCT 32 D3 T2 Wed: 15:00-15:55

LCT 33 D3 T3 Wed: 15:00-15:55

LCT 22 D3 T4 Wed: 15:00-15:55

LCT 23 D3 T5 Wed: 15:00-15:55

LCT 34 D4 T1 Wed: 15:00-15:55

LCT 35 D4 T2 Wed: 15:00-15:55

LCT 36 D4 T3 Wed: 15:00-15:55

LCT 25 D4 T4 Wed: 15:00-15:55

LCT 26 D4 T5 Wed: 15:00-15:55

Policy for Attendance

Attendance in both lecture and tutorial classes is compulsory. Students who fail to attend 80% of thelecture and tutorial classes may be awarded XX grade.

Lecture Plan

Week 1 Matrices and Gauss Elimination

Week 2 Gauss Elimination (continued),Determinants, Vector spaces, Subspaces.

Week 3 Basis, Dimension of a subspace,Row rank, Column rank, Rank in terms of determinant

Week 4 Linear transformations, Coordinate vectors,Matrix of a linear transformation

Week 5 Inner product spaces, Gram-Schmidt process,Projections and Least Squares Approximation

Week 6 Eigenvalues and eigenvectors, Algebraic and geometric multiplicities,Spectral Theorem

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Tutorial Plan

Tut 1 Matrices + Gauss Elimination

Tut 2 Gauss Elimination + Determinants + Vector spaces

Tut 3 Vector Spaces + Linear Transformations

Tut 4 Inner Product Spaces

Tut 5 Eigenvalues & Eigenvectors

Evaluation

Quiz TBA 40 Minutes 20 marks

Final Examination TBA 2 Hours 60 marks

InstructorsD1 & D3 : Balmohan V. Limaye : Room 211 E, Dept of Maths, Phone no. 7465

D2 & D4 : Murali Srinivasan (Instructor-in-charge) : Room 104 C, Dept of Maths, Phone no. 7484

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