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Linear Algebra Diagnostic topics

(latest edit: May 2021)

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; Singular value decomposition; 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.

Sample problems

Previous Linear Algebra Exams:

May 2021 

January 2021

August 2020

May 2020

August 2019

May 2019

August 2018

last updated: May 2021

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