Linear Algebra II

This course is organized as follows: In the first two lectures, we do a short review of Linear Algebra I to prepare for further developments in the subject. Then we deal with eigenvalues, eigenvectors, and diagonalization of matrices; Next, we deal with inner product spaces generalizing the main results in the plane and in the space. We use the Gram-Schmidt process to construct an orthogonal basis for a subspace of the inner space and further, give a QR decomposition of an invertible matrix. After that, we study least-squares problems and give methods for solving them. Finally, we deal with the orthogonal diagonalization problem of symmetric matrices and use the results to study quadratic forms, concentrating on positive definite quadratic forms. We also give a short discussion of the singular value decomposition of a matrix. The course gives a few applications of the main results: use the Cayley-Hamilton Theorem to find the inverse of an invertible matrix and the powers of a square matrix; use diagonalization to solve systems of linear differential equations; use the theory of projection onto subspaces to deal with Fourier series.

Each lecture contains a few in-class exercises. I will give you ten to twenty minutes to try your hands on. Afterward, I will go over the solutions with you.

Who this course is for:

• Undergraduate students who have taken Linear Algebra I