## Diagnostic Examination Information

The Mathematics Department gives a diagnostic exam each August and May on advanced calculus and linear algebra. The purpose of the exam is to determine readiness to continue in the PhD program. Any student with an assistantship or fellowship must pass the exam by the beginning of their second year in the program in order for their assistantship or fellowship to be renewed after the second year. Students who do not pass the exam are strongly encouraged to complete an MS degree in their second year and apply for jobs or to alternative PhD programs. Incoming students are strongly encouraged to take the exam this August. Students who do not take or do not pass either exam this August are required to take the corresponding undergraduate course in their first year, and will have opportunities to pass the exams in May or August.

May 2020 exams are scheduled for Wednesday, May 6 at 10:00 am (Advanced Calculus/Analysis) and 1:00 pm (Linear Algebra). Both exams will be given online and will be 2 hours long. More information will be provided as we get closer to the date.

This summer the exams will be offered August 10 at 10:30 (Advanced Calculus) and 1:30 (Linear Algebra),* room (TBA).*

### Analysis Diagnostic Examination Topics

Texts:

*Understanding Analysis*, by Stephen Abbott

*An Introduction to Analysis*, by William Wade

Math 341 Lecture Notes, by Michael Frazier

The following table shows the basic topics covered on the Analysis Diagnostic Exam in
Column 1. Column 2 shows the sections where this topic can be found in Abbot’s
*Understanding Analysis*. The third column shows where this material can be found in Wade’s
*An Introduction to Analysis*.

Topic | Understanding AnalysisStephen Abbott |
An Introduction to AnalysisWilliam Wade |
---|---|---|

logic, proofs, induction | 1.2 | 1.4 |

sets, functions | 1.2 | 1.1, 1.5-6 |

properties of ℝ, completeness | 1.3-4 | 1.2-3 |

sequences, limits of sequences | 2.2-4 | 2.1-2 |

subsequences, Cauchy sequences | 2.5-6 | 2.3-4 |

open and closed sets | 3.2 | 8.3-4* |

compact sets | 3.3 | 9.2, 9.5* |

limits of functions | 4.2 | 3.1-2 |

continuity of functions | 4.3-5 | 3.3 |

uniform continuity | 4.4 | 3.4 |

the derivative | 5.2-4 | 4.1-3 |

sequences of functions | 6.2 | 7.1 |

uniform convergence | 6.2 | 7.1 |

Riemann integration | 7.2-5 | 5.1-3 |

* Although open, closed, and compact sets are included in Wade’s text in the portion on analysis in several variables, the Diagnostic Exam will only cover these topics in one dimension.

### Linear Algebra Diagnostic Examination Topics

The following is a non-exhaustive list of typical topics covered 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; dual space, Riesz representation theorem for inner product spaces.
- Invariant subspaces; eigenspaces, triangularization and diagonalization.
- Inner products, norms; orthogonal complements and projection, minimization problems; self-adjoint and normal operators: spectral theorem. Positive operators and isometries; Polar decomposition and singular value decomposition; quadratic forms, minimizing properties of eigenvalues.
- Nilpotent operators; Jordan canonical form and real Jordan form; characteristic and minimal polynomials, Cayley-Hamilton theorem.
- Trace and determinant. Operator norm and convergence. Matrix exponential and applications.

These topics can be found in various textbooks. For example, they are covered in the following references:

- Axler's
*Linear Algebra Done Right*, Chapters 1 - 10 - R. Devaney, M. Hirsch, S. Smale: Differential Equations, Dynamical Systems, Introduction to Chaos: Chapters 5 and 6.