## 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. This summer the exams will be offered August 14 at 10:30 (Advanced Calculus) and 1:30 (Linear Algebra). 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 of 2019.

### Analysis Diagnostic Examination Topics

Texts:

*Understanding Analysis*, by Stephen Abbott

*An Introduction to Analysis*, by William Wade

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 list shows the basic topics covered on the Linear Algebra Diagnostic Exam.

- Vector spaces; subspaces; bases; spanning sets; linear independence; dimension.
- The possible types of solution sets of linear equations; linear transformations; relation between the rank and nullity of a matrix or linear transformation; determinant; trace; invertibility; transpose of a matrix.
- Characteristic polynomial; Cayley-Hamilton theorem; eigenvalues and eigenvectors; similarity; Jordan canonical form; matrices similar to a diagonal matrix.
- Inner products, inner product spaces: orthonormal bases; real symmetric matrices, orthogonal matrices; positive definite matrices. 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, students may review selected topics from the following sections of Anton’s Elementary Linear Algebra (although beware that this text does not cover Jordan canonical form or the Cayley-Hamilton theorem).

- Chapter 1 (Sections 1.1-1.8)
- Chapter 2 (Sections 2.1-2.3)
- Chapter 3 (Sections 3.1-3.3)
- Chapter 4 (Sections 4.1-4.8, 4.10)
- Chapter 5 (Sections 5.1-5.2)
- Chapter 6 (Sections 6.1-6.2)
- Chapter 7 (Sections 7.1-7.2)
- Chapter 8 (Sections 8.1-8.5)

- Axler’s Linear Algebra Done Right, Chapters 1, 2, 3, 5, 6, 8. 10
- Shilov’s Linear Algebra, Chapters 1, 2, 3, 4, 5, 6, 8, except for sections 5.6, 6.7, 6.8, 8.8.