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Course Modules for Second Year B.A.

MODULE CODE: MH4713

MODULE TITLE: Linear Algebra

PREREQUISITE MODULE(S): None

OBJECTIVES:

To present an exploration of vector spaces and matrices with particular reference to the development of computational techniques and application skills.

LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

  • be able to solve systems of linear equations using row reduction techniques and matrix operations;
  • understand the basic structural properties of vector spaces and inner product spaces and be able, for example, to determine bases and orthogonal bases for subspaces of a given vector space;
  • understand linear mappings and their relation to matrices and be able to calculate basic invariants related to such mappings;
  • be able to find the eigenvalues and eigenvectors of a matrix and use them in the process of diagonalization.

MODULE CONTENT:

  • Vectors and vector spaces; subspaces, linear independence, bases, physical applications.
  • Inner products, norm and distance, orthogonality, orthonormal basis.
  • Matrices; matrix operations, echelon matrices, algebra of square matrices, classical adjoint, matrix inversion.
  • Linear equations; methods of solution, Gaussian elimination, use of determinants.
  • Linear mappings; kernel and image, vector space isomorphisms, space of linear mappings. Linear transformations; matrix representation, change of basis.
  • Eigenvalues and eigenvectors, characteristic polynomial.

PRIME TEXT:

GROSSMAN, S.I. (1986). Multivariable Calculus, Linear Algebra and Differential Equations. Harcourt, Brace, Jovanovich.

OTHER RELEVANT TEXTS:

ANTON, H. (1973). Elementary Linear Algebra. Wiley.

KOLMAN, B. (1980). Introductory Linear Algebra with Applications. Macmillan.

MORRIS, A.O. (1983). Linear Algebra An Introduction. Van Nostrand Reinhold.

WHITELAW, T.A. (1983). An Introduction to Linear Algebra. Blackie.

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MODULE CODE: MH4763

MODULE TITLE: Calculus I: Differentiation

PREREQUISITE MODULE(S): None

OBJECTIVES:

This module is the first part of a two-semester course on differentiation and integration of functions depending on one real variable. It covers an area that is both a core part of every mathematical education, and fundamental for further studies in analysis or applied mathematics.

LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

  • differentiate functions of one real variable;
  • demonstrate knowledge and understanding of the mathematical concepts of function, limit, continuity and derivative;
  • use these concepts appropriately in solving problems and in discussing solutions.

MODULE CONTENT:

  • functions and graphs
  • slope, Newton quotient, and derivative
  • limits
  • differentiation rules for sums, products, quotients, composite functions
  • trigonometric functions, logarithms, exponential functions, and their derivatives
  • continuous functions
  • nested intervals, completeness of the real numbers
  • Intermediate Value Theorem
  • inverse functions and their derivatives

PRIME TEXT:

LANG, S. (2002). Short Calculus. Springer.

FLANDERS H. (1985). Single Variable Calculus. Freeman.

OTHER RELEVANT TEXTS:

ANTON H. (1998). Calculus. A New Horizon. Volume 1. John Wiley & Sons.

STRANG G. (1991). Calculus. Wellesley-Cambridge Press.

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MODULE CODE: MH4724

MODULE TITLE: Introduction to Statistics

PREREQUISITE MODULE(S): MH4763

OBJECTIVES:

To provide an introduction to the theory of probability and to statistical techniques in a manner which will foster understanding of concepts and development of expertise in their applications.

LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

  • be able graphically to represent sample data in an appropriate way and to correctly use summary statistics such as measures of central tendency and dispersion;
  • have an understanding of probability, random variables, probability distributions and sampling theory and be able to apply these to the analysis of sample data in order to find estimators and to test hypotheses.

MODULE CONTENT:

  • Description of sample data; graphical representation, measures of location and dispersion, mean and standard deviation. Probability theory; applications.
  • Random variables; discrete and continuous, expectation and variance.
  • Probability distributions; binomial, poisson and normal distributions.
  • Sampling theory; random sampling, sampling distributions.
  • Estimation; point and interval estimates, Student's t distribution.
  • Hypothesis testing; error types.
  • Correlation and regression; least squares, errors.
  • Testing methods; chi square test, F test, non parametric tests.

PRIME TEXT:

HOEL, P. (1976). Elementary Statistics. Wiley.

OTHER RELEVANT TEXTS:

FRANK, H. (1974). Introduction to Probability and Statistics. Wiley.

HOGG, R., CRAIG, A. (1978). Introduction to Mathematical Statistics. Collier Macmillan.

MENDENHALL, SCHAEFFER, WACKERLY (1986). Mathematical Statistics with Applications. Duxbury.

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MODULE CODE: MH4764

MODULE TITLE: Calculus II: Integration

PREREQUISITE MODULE(S): MH4763

OBJECTIVES:

This module is the second part of a two-semester course on differentiation and integration of functions depending one real variable. It covers an area that is both a core part of every mathematical education, and fundamental for further studies in analysis or applied mathematics.

LEARNING OUTCOMES:

Students who successfully complete this module should be able to:

  • integrate functions of one real variable;
  • demonstrate knowledge and understanding of the theory of differentiation and integration of such functions;
  • apply differentiation and integration techniques to solve problems (e.g. find maximum/minimum, compute area);
  • express their mathematical thoughts clearly.

MODULE CONTENT:

  • maxima and minima
  • boundedness of continuous functions on closed intervals
  • Rolle's Theorem, Mean Value Theorem
  • increasing and decreasing functions
  • antiderivatives, indefinite integrals, integration by parts, substitution
  • area, Riemann sums, definite integrals
  • least upper bound, greatest lower bound
  • Fundamental Theorem of Calculus
  • Taylor's Formula
  • infinite series, convergence

PRIME TEXT:

LANG, S. (2002). Short Calculus. Springer.

FLANDERS H. (1985). Single Variable Calculus. Freeman.

OTHER RELEVANT TEXTS:

ANTON H. (1998). Calculus. A New Horizon. John Wiley & Sons.

STRANG G. (1991). Calculus. Wellesley-Cambridge Press.

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