Linear Algebra Preliminary Examination, Format/Grading, Content Bins and Expectations of Students

 

Format/Grading

The exam will be 8 questions split into two parts. The first part of the exam will have 4 questions, one from each of the content bins discussed below. Problems in this part of the exam are generally more direct and self-contained within each of the content bins. Students are required to answer all questions in this part of the exam.

The second part of the exam will have 4 questions covering more advanced material or more integrated material within or across the content bins. Students are required to choose 2 of the 4 questions to answer in this part of the exam.

The grading of the exam and determination of pass/fail is based upon student responses for the six problems submitted by the student. Solutions (proofs, examples/counterexamples, etc.) are graded on both correctness of the mathematical argument and correct writing. Extraneous material provided by the student that is not relevant to a solution negatively impacts the assessment of the solution. Progress towards the problem as stated is required for partial credit; solutions of a similar related or a partial problem are considered not relevant. Incorrect or imprecise notation, or lack of proper mathematical proof writing negatively impacts the assessment of the solution.

Content Bins

  1. Vector Spaces: examples of vector spaces, linear independence, span, basis vectors, rank, finite-dimensional and infinite-dimensional vector spaces, subspace, sum, direct sum. (See Chapters 1, 2 3 and 4 of Axler). Linear Transformations: matrix representation, change of basis, products of linear maps, null space, range, fundamental subspaces, injectivity, surjectivity, rank-nullity theorem, invertibility, isomorphism, operators. (See Chapters 3 and 10 of Axler).
  2. Inner Product Spaces: inner products, vector norms, matrix norms, norms induced by inner products, orthogonality, orthonormality, Gram-Schmidt procedure, orthogonal complements, orthogonal projection, Cauchy-Schwartz inequality. (See Chapter 6 of Axler).
  3. Classes of Operators and Matrices: self-adjoint, Hermitian, symmetric, normal, unitary, positive definite, isometry, nilpotent, spectral theorem. (See Chapters 7, 8 and 9 of Axler).
  4. Eigenvalues and Eigenvectors: characteristic polynomial, eigenvalues/eigenvectors, generalized eigenvectors/eigenspaces, multiplicity, trace, determinant, similarity transformation, diagonalization, determinant, singular value decomposition. (See Chapter 5, 7, 8 and 10 of Axler). For determinants, see Chapter 10 of Axler. Singular value decomposition: (See Chapter 7 of Axler). Canonical Forms: Jordan Form, minimal polynomial, Cayley-Hamilton theorem. (See Chapters 8 and 9 of Axler).

The exam assumes students have competence with basic skills and concepts covered in undergraduate linear algebra courses. These include solving linear systems of equations, Gaussian elimination, row echelon form, LU-decomposition, linear independence, matrix multiplication, matrix inverses, determinants, properties of R^n, general vector spaces and subspaces, computing bases for fundamental subspaces of a matrix, eigenvalues/eigenvectors, diagonalizability, eigendecomposition of symmetric/Hermitian matrices and nonsymmetric/non-Hermitian matrices, eigendecomposition of self-adjoint matrices using an orthogonal similarity transformation, inner products, Gram-Schmidt algorithm, least squares, singular value decomposition. Such skills may be required to work problems in any of the content bins.

General expectations of students

Students should be able to write clear, correct, and concise proofs based on definitions and to use standard techniques where applicable. Students should be able to prove standard results in each of the content bins. Students should also be able to provide simple, straightforward counterexamples to false statements (e.g., by the removal of a necessary condition in a standard result), and students are expected to justify their counterexamples.

References:

  1. Sheldon Axler: Linear Algebra Done Right
  2. David Cherney, Tom Denton, Rohit Thomas and Andrew Waldron: Linear Algebra. https://www.math.ucdavis.edu/~linear/