Linear Algebra & Applications

Upon completion of this course the student will be able to:

CO1: Interpret existence and uniqueness of solutions to linear equations, linear dependence and independence of vectors.

CO2: Characterize linear transforms using the concepts of existence and uniqueness.

CO3: Construct a spanning set/basis for i) a vector space or subspace ii) Range and kernel of transformation

CO4: Apply Gram-Schmidt Process to construct an orthogonal basis for a subspace.

CO5: Evaluate the best possible solution for an inconsistent system and a regression model with least square error

CO6: Apply mathematical tools such as PCA and SVD for dimensional reduction (feature extraction) and image processing applications

Unit 1

System of Linear Equations, Row Reduction and Echelon Forms, Vector Equations, Linear combinations, Matrix equation Ax=b, Solution sets of linear systems, Linear Independence.

Unit 2

Linear Transformation/mapping, Matrix of linear transformation, one-to-one, Onto mapping, Isomorphism, Inverse of a matrix, Invertible matrix theorem, Determinant, determinant as area & volume.

Unit 3

Vector Spaces and subspaces, null spaces, column spaces, linearly independent sets: Bases, Spanning Set, Dimension of a vector space, rank. Kernel and Image of Linear Transformation, Singular and Non-singular transformation

Unit 4

Eigenvalues & Eigenvectors, The Characteristic Equation, Diagonalization, Inner product, outer product, length and orthogonality, orthogonal sets, orthogonal projections, Gram-Schmidt Process, Least square problems, Least-square lines.

Unit 5

Diagonalization of symmetric matrices, Quadratic forms, constrained optimization, Singular value decomposition, Principal Component Analysis: Application to image processing.

Text book

David C. Lay

Linear Algebra and its Application, Pearson Education, Fifth Edition. 2016

Reference Books

1 Seymour Lipschutz, Schaum's Outline of Linear Algebra, Sixth Edition. 2017.

2) Strang, G. Linear Algebra and its Applications, Third Edition, Thomson Learning.

3) Kenneth Hoffman and Ray Kunze, Linear Algebra, 2^nd edition, Pearson Education (Asia) Pte. Ltd. Prentice Hall of India, 2004.