Course Identification

• Some familiarity with elementary linear algebra would be helpful, though not strictly necessary, since most of the basic concepts will be reviewed (quickly).
• Elementary concepts of calculus (differentiation, integration, continuity) will enter in the second part of the course.

This course will both review and extend a number of basic mathematical tools which are generally useful in applications and are typically assumed as prerequisites for many of the current courses. The lectures will focus on the following topics:

1. Linear Algebra: Jordan forms. Special classes of matrices (unitary, Hermitian, normal, positive definite, stochastic). Singular value decompositions. Pseudoinverses. Linear systems of equations.
2. Normed linear spaces: Norms. Basic inequalities. Inner product spaces. Orthogonal systems. Projections, Orthogonal projections, Mean square approximation.
3. Differential and difference equations: Systems of first order differential equations. Higher order scalar equations.
4. Matrix valued functions: Mean value theorems. Fixed point theorems. The inverse function theorem. The implicit function theorem. Newton's method.
5. Optimization: Extremal problems. Lagrange multipliers. Convexity.
6. Matrices with nonnegative entries: Perron-Frobenius theorem. Birkhoff-von Neumann theorem. von Neumann's inequality, Ky Fan's inequality.

Upon successful completion of this course students should be able to:

1. Demonstrate familiarity with the basic tools of linear algebra with special emphasis placed on the ability to compute.
2. Calculate Jordan forms.
3. Solve systems of linear equations.
4. Compute projections (both orthogonal and skew), best mean square approximation and applications of singular value decomposition.
5. Apply the acquired knowledge to multivariable calculus (including implicit function theorems and extremal problems with constraints).

Lectures will be adapted from selected parts of the text: Linear Algebra in Action. Copies of this text are available in the Math Library and the library adjacent to the Ulmann building.