By the end of this course, students will be able to:

1. Identify and apply the fundamental properties of vector spaces, linear transformations, and inner product spaces in engineering contexts.

2. Perform advanced matrix operations, including eigenvalue analysis and matrix factorizations, to model and solve linear systems arising in engineering problems.

3. Formulate and manipulate tensor fields in various coordinate systems, and apply tensor calculus to represent physical quantities such as stress and strain.

4. Analyze and solve problems involving complex-valued functions using techniques from complex analysis, including contour integration and residue calculus.

5. Use the principles of calculus of variations to derive governing equations for dynamical systems and optimize engineering functionals.