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Course code  :  PMAT 21263

Title               :  Linear Algebra

Pre-requisites :  PMAT 11232

 Learning outcomes:

Upon successful completion of the course students should be able to

1. demonstrate understanding of the concepts of vector space, subspace linear independence, span and basis
2. determine eigenvalues and eigenvectors and solve eigenvalue problems
3. describe algebraic and geometric multiplicities of eigenvalues and linearly independent eigenvectors
4. apply principles of matrix algebra to linear transformations
5. demonstrate an understanding of inner products and associated norms

 Course Contents:

Vector Spaces: Vector Spaces, Subspaces, Spanning Sets and Linear Independence, Basis and Dimension,
Extension Theorem, Coordinates, Change of Basis and Transition Matrix, Similarity, Dimensional Theorem.

Linear transformations: Linear Transformation, Kernel and Range of Linear Transformation, Rank and Nullity
Theorem, Isomorphisms, Matrix Representation of Linear Transformation, Applications of Linear Transformation.

Eigenvalues and Eigenvectors: Characteristic Polynomial, Eigenvalues and Eigenvectors, Eigen Spaces,
Diagonalization, Inner Product Spaces, Gram-Schmidt Orthogonalization Process, Orthogonal Complement,
Orthogonal Projections, Cayley-Hamilton Theorem, Minimum Polynomial of Matrices of Order Three

 Method of Teaching and Learning: A combination of lectures and tutorial discussions.

 Assessment: Based on tutorials, tests and end of course examination.

 Recommended readings:

1. Larson, R. & Falvo, D.C. (2016). Elementary Linear Algebra, Brooks Cole.
2. Andrilli, S. & Hecker, D. (2016). Elementary Linear Algebra, Elsevier Science.
3. DeFranza, J. & Gagliardi, D. (2015). Introduction to Linear Algebra with Applications, Waveland Press.
4. Lay, D.C., Lay, S.R. & McDonald, J.J. (5th Ed., 2015). Linear Algebra and Its Applications, Pearson

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Course code  :  PMAT 12212

Title               :  Mathematics for Computing II

Pre-requisites :  PMAT 11212

Learning Outcomes:

Upon successful completion of the course unit, the student should be able to:

1. demonstrate knowledge in matrices
2. solve systems of linear equations
3. compute eigenvectors and eigenvalues
4. solve real world problems using counting techniques
5. demonstrate knowledge in basic graph theory.

Course Contents:

Matrices - Matrix notations, Algebra of matrices, Inverse of a matrix, Elementary row operations and
calculation of inverse matrix, Determinant of a matrix, Minor and cofactor, Properties of determinants, Adjoint
of a matrix and calculation of inverse matrix.
Systems of Linear Equations - Matrix form of a system of equations and conditions for unique solutions,
augmented matrix and echelon form, Conditions for existence of a unique solution, infinitely many solutions
and no-solution.
Vector Spaces- Linear Independence, Basis and Rank, Linear Mappings, Eigenvalues and Eigenvectors
Counting Techniques - Factorial notation, Sum rule and product rule, Pigeonhole principle, Permutations and
combinations, Binomial expansion.
Graphs - Graph notation and basic definitions, Complete and bipartite graphs, Euler and Hamiltonian circuits,
Directed and undirected graphs, Planar graphs, Graph colouring, Trees and their basic properties

Method of Teaching and Learning: Lectures, interactive classroom sessions, and case discussions 

Assessment: End of course unit examination, group assignment, mid-term examination, class attendance

Recommended Reading:

1. Johnsonbaugh, R. (8th Ed., 2017). Discrete Mathematics, Pearson.
2. Rosen, K.H. & Krithivasan, K. (7th Ed., 2011). Discrete Mathematics and Its Applications,
McGraw-Hill.
3. Kreyzig, E. (8th Ed., 2006). Advanced Engineering Mathematics, Wiley Student Edition.
4. Susanna, S. E. (2010). Discrete Mathematics with Applications, Cengage Learning.
5. Larson, R. & Falvo, D. (6th Ed., 2009). Elementary Linear Algebra, Houghton Mifflin Harcout
Publishing Compan

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Course code   : PMAT 12203

Title               : Introduction to Calculus

 Learning Outcomes:

At the end of this course, the student should be able to

1. define the basic concepts of limits
2. evaluate the limits of functions
3. find rates of change
4. recognize the continuity of a function
5. explain basic rules of differentiation
6. identify indeterminate forms and methods of integration
7. define an ordinary differential equation
8. solve applications related to the ordinary differential equations appear in biology

Course Contents:

Limits and Derivatives: Intuitive definition of a limit, one sided limits, Calculating limits using limit laws,
Derivatives and Rates of change, Derivative as a function
Continuity: Continuity at a point, continuity on an interval, Intermediate value theorem
Differentiation Rules: Derivatives of polynomials and Exponential functions, Product and quotient rules,
Derivatives of Trigonometric functions, Chain rule, Implicit differentiation, Derivatives of Logarithmic functions
Applications of Differentiation: Maximum and Minimum values, Indeterminate forms and l’Hospital’s rule,
Taylor's formula, Newton's method
Techniques of Integration: Basic integrals, Integration by parts, Trigonometric integrals, Trigonometric
substitution, Integration of rational functions by partial fractions, Improper integrals
Introduction to Differential Equations: Separable equations, Linear equations, Applications related to biology

Method of Teaching and Learning: A combination of lectures and tutorial discussions.

Assessment: Based on the tutorials, tests, and end of course examination.

Recommended reading: 

1. Ayres, Jr .F. & Mendelson, E., (6^th edition, 2012). Schaum's Outline of Calculus, McGraw-Hill.
2. Arora, S. & Malik, S.C., (5^th edition, 2017). Mathematical Analysis, New Age International.
3. Stewart, J., Clegg, D.K., & Watson, S. (11th Ed., 2020). Calculus: early transcendentals. Cengage
   Learning.

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Course code  :  PMAT 12253

Title               :  Theory of Calculus

Pre-requisites :  PMAT 11223

Learning Outcomes:

On successful completion of the course, the student should be able to;

1. classify a real number as a natural, whole, integer, rational, or irrational number and demonstrate knowledge
of the axiomatic description of the field of real numbers and prove theorems from the given set of axioms
2. evaluate limits given analytic, graphical, numerical function information and describe in simple language the
statements of limit laws and use these laws to evaluate limits and state the definition of continuity and use
the definition to ascertain the continuity or dis-continuity of a function at a point
3. state the limit definition of derivative of a function at a point and use the limit definition to calculate a
derivative or identify where the derivative fails to exist at a point
4. apply the Chain Rule to find the derivative of a composition of functions and interpret both continuous and
differentiable functions geometrically and analytically and apply Rolle's Theorem, the Mean Value Theorem
5. explain indeterminant forms and use L’Hopital’s rule to evaluate limits involving indeterminant forms
6. describe upper and lower Darboux sums, Riemann sum and compare them and write the statement of the
Fundamental Theorems of Calculus and explain what the theorems say about definite integrals
7. visualize and sketch the surface generated by revolving a graph of a function about an axis of evolution and
calculate the volume of a solid and volume of a solid revolution by using disk, washer and cylindrical
shells methods
8. classify improper integrals and determine the integrals, identify the types of improper integrals and rewrite
them as proper integrals with a limits and use the words convergent and divergent to describe an
improper integral.

 Course Contents:
Real Numbers: Field Structure, Ordered fields and their properties, Open and Closed sets, Maximum, Minimum,
Supremum and Infimum of a set, Completeness axioms, Archimedean Property, Denseness of subsets of real
numbers
Real Valued Functions of a Real Variable: Review of Polynomial, Rational, Algebraic, Trigonometric,
exponential and Logarithmic Functions, Composite Functions, Piece-wise Functions
Functions and Limits: Limits of Functions, Left and Right-hand limits, Squeeze theorem, Continuous Functions,
Asymptotes and limits involving infinity
Derivative and Applications: Derivative of a function, Chain rule, Logarithmic and Implicit differentiation,
Higher order derivatives, Rolle’s Theorem, Mean Value Theorem
Indeterminate Forms: L’hospital Rule.
Integrals: Darboux and Riemann Integrations, Fundamental Theorem of Calculus, Improper Integrals (First &
second kind)
Applications of Integrals: Volumes and Solid revolutions, Volumes by Cylindrical Shells, Mean Value Theorem
for Integrals, Arc Length, Area of a Surface of Revolution

 Method of Teaching and Learning: A combination of lectures and tutorial discussions.

 Assessment: Based on tutorials, tests and end of course examination.

 Recommended Reading:

1. Stewart, J. (9th Ed., 2020). Calculus Early Transcendentals, Cengage Learning, Inc.
2. Larson, R. & Edwards, B.H. (11th Ed., 2018). Calculus, Brooks/Cole, Cengage Learning
3. Ross, K.S. (2nd Ed., 2015). Elementary Analysis: The Theory of Calculus, Springer.
4. Hass, J., Heil, C. & Weir, M.D. (14th Ed., 2017). Thomas' Calculus: Early Transcendentals, Single
Variable, Pearson