Course

Unit 1

Vector Spaces: Vector spaces – Sub spaces – Linear independence – Basis – Dimension. (8 hrs)

Inner Product Spaces: Inner products – Orthogonality – Orthogonal basis – Gram Schmidt Process – Change of basis – Orthogonal complements – Projection on subspace – Least Square Principle. QR- Decomposition.

Unit 2

Linear Transformations: Linear transformation – Relation between matrices and linear transformations – Kernel and range of a linear transformation – Change of basis – Nilpotent transformations. Symmetric and Skew Symmetric Matrices, Adjoint and Hermitian Adjoint of a Matrix, Hermitian, Unitary and Normal Transformations, Self Adjoint and Normal Transformations.

Unit 3

Eigen values and Eigen vectors: Eigen Values and Eigen Vectors, Diagonalization, Orthogonal Diagonalization, Quadratic Forms, Diagonalizing Quadratic Forms, Conic Sections.  Similarity of linear transformations – Diagonalisation and its applications – Jordan form and rational canonical form.

Case Studies: Applications on least square and image transformations.

Lab Practice Problems: Matrices, Matrix operations. Solving system of linear equations, rank and nullity. Orthogonality. Matrix of linear transformations. Affine transformations, scaling, shifting and rotation of images. Eigen values and eigen vectors and matrix decompositions.

Course Objectives

• Understand the basic concepts of vector space, subspace, basis and
• Familiar the inner product Finding the orthogonal vectors using inner product.
• Understand and apply linear transform for various matrix
• Familiarize the concepts of eigenvalues and eigenvectors and its

Course Outcomes

CO1: To Understand the basic concepts of vector space, subspace, basis and dimension.

CO2: To Understand the basic concepts of inner product space, norm, angle, Orthogonality and projection and implementing the Gram- Schmidt process, to obtain least square solution

CO3: To Understand the concept of linear transformations, the relation between matrices and linear transformations, kernel, range and apply it to change the basis and to transform the given matrix to diagonal matrix.

CO4: To understand the eigen values and eigen vectors and apply to transformation problems.

CO5: To perform case studies on least square and image transformations.

CO-PO Mapping

Evaluation Pattern

• CA – Can be Quizzes, Assignment, Lab Practice, Projects, and Reports

Text books

1. Howard Anton and Chris Rorres, “Elementary Linear Algebra”, Tenth Edition, John Wiley & Sons,

References

1. Nabil Nassif, Jocelyne Erhel, Bernard Philippe, Introduction to Computational Linear Algebra, CRC press,
2. Sheldon Axler, Linear Algebra Done Right, Springer,
3. Gilbert Strang, “Linear Algebra for Learning Data”, Cambridge press,
4. Kenneth Hoffmann and Ray Kunze, Linear Algebra, Second Edition, Prentice Hall,
5. Mike Cohen, Practical Linear Algebra for Data Science, Oreilly Publisher,