Course Code:                 MATH 2273

Course Title:                  Linear Algebra I

Course Type:                 Core

Level:                             2

Semester:                       1

No. of Credits:                 3

Pre-requisite(s):              MATH 1141 – Introduction Linear Algebra and Analytic

Geometry.

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 pre-requisite.

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 in-course 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 Gram-Schmidt 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 Rank-Nullity Theorem. Matrix of a linear transformation. Composition of transformations. Change of basis.
• Systems of Linear Equations: Homogeneous and Non-homogeneous systems. Augmented Matrix. Row space and Column Space of a Matrix. Elementary row and column transformations: Reduced Row-Echelon 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 Gram-Schmidt Orthogonalisation Process. Orthogonal matrices.
• Eigenvalues and Eigenvectors: The properties of eigenvalues and eigenvectors. Diagonalisation of Matrices. Similarity. The Characteristic Polynomial. The Cayley-Hamilton Theorem.

Assessment

Coursework - Two coursework exams (20% each), Assignments (10%).   Total: 50%

Final Examination:  One 2-hour 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

Essential Texts:

Linear Algebra and Matrix Theory – Jimmie & Linda Gilbert, Academic Press Inc, (1995).

Linear Algebra – Serge Lang. Springer-Verlag. 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