# MS Basic Exam for Advanced Calculus and Linear Algebra

## M.S. Basic Exam for Advanced Calculus and Linear Algebra

#### About the M.S. Basic Exam

Each exam consists of a 2-hour written exam, and a take-home exam.

#### Schedule

The exam will be given prior to the start of each Fall semester. The take-home part will be due several days later. Results and recommendations will be provided prior to the end of the add/drop period. .

#### Content of the M.S. Basic Exam

Topics to be covered in the M.S. Basic Exam Advanced Calculus include the following:

• Elementary properties of Open/Closed/Compact/Connected sets in R^n
• Numerical sequences and series
• Limits, Cauchy sequences, and convergence
• Continuity
• Continuity and compactness/connectedness
• Uniform continuity
• Sequences and series of functions; uniform convergence
• Calculus of real-valued functions: Differentiation, mean value theorems, Taylor's theorem
• Definition and existence of the Riemann integral
• Fundamental Theorem of Calculus
• Integration and differentiation of series/sequences of functions

Topics to be covered in the M.S. Linear Algebra Exam include the following:

• Vector Spaces
• Linear Independence
• Basis
• Dimension
• Linear Transformation
• Matrix Representations
• Rank
• Range Space
• Null Space
• Eigenvalues and eigenvectors
• Diagonalizations
• Canonical forms
• Inner product spaces
• Orthogonal basis
• Symmetric and Hermitian matrices and properties

#### Grading of the M.S. Basic Exam

The examination committees will send the graduate program committee their course recommendations within a 7 day period after the written exam is conducted. These may include advanced Calculus Math 451, or real analysis Math 551 and/or linear algebra Math 343, Math 441, Math 543. The recommendation will be based on the student's background and performance on the exam.