Linear Algebra - MA25C02

MA25C02: 

Linear Algebra

 

Course Objectives:

• To impart foundational knowledge in linear algebra essential for analysing and solving problems in engineering applications.

• To provide the knowledge on computation using software and interpret key linear algebra concepts using software.

 

Vector Spaces Introduction to Vector Spaces, Examples, Subspaces, Linear Combinations, Span, Generating Sets, Linear Dependence and Independence, Basis and Dimension, Dimension of Subspaces.

Activities: Open-Source software, exercises to test linear dependence and independence using rank, compute span and basis of a set of vectors, determine the dimension of subspaces, and illustrate the concept of subspace and basis in 𝑹^𝟐/𝑹^𝟑 with visualization.

Linear Transformations and Diagonalization: Null space, Range, Dimension Theorem (statement only), Matrix representation of a linear transformation, Eigenvalues & Eigenvectors, Diagonalizability.

Activities: Open-Source software, exercises to compute the matrix representation of a linear transformation, find the null space and range of a matrix, and compute eigenvalues and eigenvectors of a matrix.

Inner Product Spaces:Inner product, Norms, Cauchy, Schwarz inequality, Gram, Schmidt orthogonalization, Simple problems (up to 𝑹^𝟑).

Activities: Open-Source software, exercises to compute inner products and vector norms.

 

Matrix Decomposition: Orthogonal transformation of a symmetric matrix to diagonal form

- Positive definite matrices, QR decomposition, Singular Value Decomposition (SVD), Least squares solutions- simple problems (up to 3 × 3 matrices )

Activities: Open-Source software, exercises to check if a matrix is positive definite, perform QR decomposition and SVD using built-in functions.

 

Weightage: Continuous Assessment: 40%, End Semester Examinations: 60%.

Methodology for Continuous Assessment: Assignment (20%), Software activity (20%), Quiz (20%), Internal Examinations (40%).

 

References:

1. Friedberg, S. H., Insel, A. J., & Spence, L. E. (2022). Linear algebra. Pearson.

2. Lay, D. C., Lay, S. R., & McDonald, J. J. (2020). Linear algebra and its applications with MATLAB. Pearson.

3. Bronson, R. (2011). Schaum’s outline of matrix operations. McGraw-Hill Education.

4. Strang,  G.,  &  Thomson,  R.  (2005).  Linear  algebra  and  its  applications. Brooks/Cole.

5. Lipschutz, S., & Lipson, M. (2009). Schaum's outline of linear algebra. McGraw- Hill.

6. Kreyszig, E. (2018). Advanced engineering mathematics. Wiley India.