# MS Basic Exam for Advanced Calculus and Linear Algebra

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## 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.

#### Textbooks

Advanced Calculus

- Elementary Analysis: The Theory of Calculus by Kenneth Ross (Used for Math 451)
- Principles of Mathematical Analysis by Rudin (A standard Advanced Calculus Text)

Linear Algebra

- Elementary Linear Algebra by Kolman (Used for Undergraduate Linear Algebra)
- Introduction to Linear Algebra by Strang (Used for Applied Linear Algebra)
- Linaer Algebra by Hoffman and Kunze (Used for Graduate Linear Algebra)

Please send any questions or comments to gradprog@math.wvu.edu