Course Code: MATH 2273
Course Title: Linear Algebra I
Course Type: Core
Level: 2
Semester: 1
No. of Credits: 3
Prerequisite(s): MATH 1141
Course Rationale
The study of linear algebra is motivated by the geometry of problems in two and three dimensions. A clear understanding of the concepts of linear algebra is essential for the proper description and representation of all physical and mathematical phenomena in higher dimensions. The algorithms of linear algebra are also central to the theory of scientific computing and numerical analysis.
A first course in linear algebra serves as an introduction to the development of logical structure, deductive reasoning and mathematics as a language. For students, the tools developed from a course in linear algebra will be as fundamental in their professional work as the basic tools of calculus. For these reasons, this course is a core course for students pursuing a major in mathematics.
Course Description
Students who take this course will require a solid grounding in set theory and basic logic. For this reason, MATH 1140 is listed as a prerequisite.
The course begins with a study of abstract linear algebra which involves vector spacers and linear transformations. Formulating such an approach leads to a study of linear equations and the technique of elementary row transformations used for solving them. The concepts of rank and equivalence are introduced. Determinants are discussed in terms of permutations. The important concepts of orthogonality, eigenvalues, eigenvectors are studied. A treatise on quadratic forms, diagonalisation of matrices and the Cayley – Hamilton theorem is included. The writing of detailed proofs is incorporated throughout.
Assessment is designed to encourage students to work continuously with the course materials. Active learning will be achieved through marked assignments supplemented bu (unmarked) problem papers, allowing continuous feedback and guidance on problem solving techniques in tutorials and lectures. Assessment will be based on the marked assignments and incourse tests followed by a final examination based on the whole course
Learning Outcomes
Upon successful completion of the course, students will be able to:
 Describe linear systems in terms of vectors and matrices.
 Perform dimension, rank, nullity and linear transformations.
 Explain the basic concepts of vector spaces and determinants.
 Express the product of two matrices in terms of elementary matrices.
 Describe the properties of eigenvalues and eigenvectors.
 Investigate the GramSchmidt process and how it affects orthogonality.
 Diagonalise matrices.
 Explain the Cayley – Hamilton theorem with respect to a square matrix.
 Use computer software for solving linear algebra problems.
Course Content
 Introduction of Vector Spaces: The definition of a vector space over an arbitrary field. Examples of vector spaces. Preliminary results.
 Subspaces: Definition and examples. Intersections of and Direct Sums of subspaces.
 Linear Independence and Bases: Linear combinations. Linear Span. Linear Independence. Bases. Dimension. Examples of vector spaces of finite dimension and of infinite dimension. The dimension of a subspace.
 Linear Transformations: Definition and resulting properties. Null Space and Range. Rank. The RankNullity Theorem. Matrix of a linear transformation. Composition of transformations. Change of basis.
 Systems of Linear Equations: Homogeneous and Nonhomogeneous systems. Augmented Matrix. Row space and Column Space of a Matrix. Elementary row and column transformations: Reduced RowEchelon form. Elementary Matrices. Matrix products via elementary row transformations. Matrix products expressed as products of elementary matrices.
 Determinants: Permutations. Definition of Determinant. Properties of Determinant. Cramer’s Rule. Cofactors and the inductive definition of Determinant. Determinants and Inverses of Matrices.
 Inner Product Spaces: Properties of Inner Products. Orthogonality. Norms. Orthonormal bases. The GramSchmidt Orthogonalisation Process. Orthogonal matrices.
 Eigenvalues and Eigenvectors: The properties of eigenvalues and eigenvectors. Diagonalisation of Matrices. Similarity. The Characteristic Polynomial. The CayleyHamilton Theorem.
Assessment
Coursework  Two coursework exams (20% each), Assignments (10%). Total: 50%
Final Examination: One 2hour written paper. Total: 50%
Course Calendar
WEEK 
LECTURE SUBJECTS 
TUTORIALS / COURSEWORK EXAMINATIONS 
1 
Introduction and Course Overview Abstract Vector Spaces and Subspaces 
Assignment # 1 given 
2 
Linear dependence and bases


3 
Linear transformation and their matrices. 
Assignment # 1 due Assignment # 2 given 
4 
Elementary row transformation and elementary matrices . 

5 
Row equivalence and rank. 
Assignment #2 due Assignment #3 given 
6 
Solutions of systems of linear equations. 
First coursework examination

7 
Determinants. 
Assignment # 3 due Assignment #4 given 
8 
Inner Products Spaces


9 
Gram – Schmidt Orthogonalisation.

Assignment # 4 due Assignment # 5 given 
10 
Characteristic Polynomial


11 
Similarity, diagonalisation 
Assignment # 5 due Assignment # 6 given 
12 
Cayley – Hamilton theorem 
Second coursework examination

13 
Revision 
Assignment # 6 due

Required Reading
Essential Texts:
Linear Algebra and Matrix Theory – Jimmie & Linda Gilbert, Academic Press Inc, (1995).
Linear Algebra – Serge Lang. SpringerVerlag. Third Edition, 1987.
Solved Problems in Abstract Algebra – Edward Farrell, Department of Mathematics & Computer Science, (2009).
Other Reference Texts:
Linear Algebra by D.C.Murdoch. Wiley, ISBN 10 – 0471625000., (1970).
Linear Algebra with Applications by LEON, Steven J.
Linear Algebra with Applications by WILLIAMS, Gareth