Course abstract

The course will assume basic knowledge of class XII algebra and a familiarity with calculus. Even though, the course will start with defining matrices and operations associated with it. This will lead to the study of system of linear equations, elementary matrices, invertible matrices, the row-reduced echelon form of a matrix and a few equivalent conditions for a square matrix to be invertible. From here, we will go into the axiomatic definition of vector spaces over real and complex numbers, try to understand linear combination, linear span, linear independence and linear dependence and hopefully understand the basis of a finite dimensional vector space. We will then go into functions from one vector space to another, commonly known as linear transformations. For the finite dimensional case, we will see that all functions can be understood through matrices and vice-versa. We will then define inner/dot product in a vector space. This leads to the understanding of length of a vector and orthogonality between vectors. As our main result in this part, we will understand the Gram-Schmidt orthogonalization process. Finally, we will go into the topic of eigenvalues and eigenvectors associated with a square matrices or linear operators. As a final result, we will learn the spectral theorem for Hermitian/Self-adjoint matrices. As an application, we will classify the quadrics.