The following is a non-exhaustive list of topics included on the Linear Algebra Diagnostic Exam.
- Vector spaces over real and complex numbers; subspaces; bases; spanning sets; linear independence; dimension.
- Linear transformations; rank and nullity; matrices, change of basis formula, similarity.
- Eigenspaces, diagonalization.
- Inner products, norms; orthogonal complements and projection, Riesz representation theorem, minimization problems;
- self-adjoint and normal operators: spectral theorem. Unitary operators and isometries; minimizing properties of eigenvalues.
- Jordan canonical form; characteristic and minimal polynomials, Cayley-Hamilton theorem.
- Trace and determinant.
It is important for students to have a conceptual understanding of the material and to have a good grasp of proof techniques.
These topics can be found in various textbooks. For example, they are covered in the following reference:
Linear Algebra Done Right, by Sheldon Axler, 3rd. edition (2015). Springer-Verlag, Undergraduate Texts in Mathematics.
Previous Linear Algebra Exams: