The purpose of the preliminary exams in the Department of Mathematics and Statistical Sciences at UCD is to evaluate readiness for doctoral study. To this end, the exam primarily tests material typically studied at the advanced undergraduate level and beginning graduate level, but with questions posed at a level to test maturity of understanding. Since doctoral study requires extensive independent study and research, adequate preparation for the exam also requires independent mastery of some material which does not typically appear in every undergraduate (or beginning graduate) level course on the topic. Consequently, no graduate course provides the entire content material for this exam. The questions on the exam will generally be at the level of the exercises in the references.

1. Systems of Equations: Gaussian elimination, row-echelon forms, LU-decomposition, inverse of a matrix

See Chapter 1 of Strang (LAA), Chapter 2 of Strang (ILA), or Chapters 1 and 2 of Lay.

2. Practical Solution of Linear Systems: partial pivoting, condition number, ill-conditioned systems

See Chapters 3.8 and 6.4 of Nobel and Daniel, Chapter 7.1 and 7.2 of Strang (LAA), or Chapter 9 of Strang (ILA)

3. Vector Spaces: examples, fundamental subspaces of a matrix

See Chapter 2 of Strang (LAA), Chapter 3 and 4.1 of Strang (ILA), Chapter 4 of Lay, or Chapter 1 of Axler.

4. Dimension Theory: linear independence, basis vectors, rank

See Chapter 2.1-2.3 of Strang (LAA), Chapter 3.4 of Strang (ILA), Chapters 1.6 and 4.3-4.6 of Lay, or Chapter 2 of Axler.

5. Linear Transformations: matrix representation, change of basis, Cayley-Hamilton theorem

See Chapters 3 and 7-9 of Axler, Chapters 6 and 9.2 of Nobel and Daniel, or Chapter 7 of Strang (ILA) and Chapter 2.4 of Horn and Johnson.

6. Eigenvalues and Eigenvectors: characteristic polynomial, similarity, determinants

See Chapters 5, 8, and 9 of Axler, Chapter 5 of Strang (LAA), Chapter 1 of Horn and Johnson, Chapter 6 of Strang (ILA), or Chapters 5.1-5.5 and 7.1 of Lay. For determinants, see Chapter 4 of Strang (LAA), Chapter 5 of Strang (ILA), Chapter 10 of Axler, or Chapter 3 of Lay.

7. Inner Product Spaces: vector and matrix norms, inner products

See Chapters 6 and 7 of Axler, Chapter 5.6-5.9 in Nobel and Daniel or Chapters 3.1-3.4, 3.6, and 5.5 of Strang (LAA). Some additional information can be found in Chapter 5 of Horn and Johnson.

8. Classes of Matrices: Hermitian or Self-adjoint, symmetric, normal, unitary, positive definite

Extensive information on these topics can be found in Chapter 7 of Axler and Chapters 2, 4, and 7 of Horn and Johnson. A particularly nice treatment of Positive Definite Matrices can be found in Strang (LAA). Chapter 4.4 and 4.6 of Strang (LAA) are also good reading.

9. Matrix Decompositions: Schur, QR, singular value, polar, least squares and generalized inverses

These topics are covered in Chapters 5.8-5.9, 7.5, and 8 of Nobel and Daniel. Also see Chapters 3.3, 3.4, and 7.3 and Appendix A of Strang (LAA) and Chapter 2.3-2.6 of Horn and Johnson. A more theoretical approach can be found at the end of Chapter 7 of Axler.

10. Canonical Forms: Jordan, minimal polynomial

It helps to look at more than one approach to this problem. See Chapter 3 of Horn and Johnson, Appendix B of Strang (LAA), Chapter 9 of Nobel and Daniel, and Chapter 8 of Axler.

## References:

1. Sheldon Axler: Linear Algebra Done Right
2. Ben Nobel, James Daniel: Applied Linear Algebra
3. Roger Horn, Charles Johnson: Matrix Analysis
4. Gilbert Strang: Linear Algebra and Its Applications (LAA)
5. David Lay: Linear Algebra and Its Applications (undergraduate text)
6. Gilbert Strang: Introduction to Linear Algebra (ILA) (undergraduate text)