Course Title: Introductory Linear Algebra and Analytical Geometry
Level: 1
Semester: 2
No. of Credits: 3
Pre-requisite(s): Units 1 and 2 of CAPE Pure Mathematics or equivalent.
Course Rationale
Motivated by geometry of two and three dimensions, linear algebra is the easiest context in which a theory of great utility and beauty can be developed. A clear understanding of the concepts of linear algebra is essential to the understanding of all physical and mathematical phenomena in higher dimensions. The algorithms of linear algebra are at the centre of much of scientific computing. A first course in linear algebra and analytical geometry involving vectors also serves as an introduction to the development of logical structure, deductive reasoning and mathematics as a language.
For students the tools of linear algebra and analytical geometry will be as fundamental in their professional work as the tools of calculus.
Course Description
This course can be divided into two sections of (i) vectors and (ii) matrices.
In (i) we begin with the algebraic definition of a vector as elements of a vector space. We proceed to the geometric interpretation of a vector and define the norm of a vector. In this way unit vectors and the Triangle Inequality are described. The dot product is defined for vectors and applications such as work done by a force in moving an object is given. We can now speak of parallel and perpendicular vectors geometrically.
Complex numbers are explained in terms of vectors. The geometrical interpretation of these is seen in the Argand diagram. Also roots of polynomials are evaluated by de Moivre’s theorem and illustrated in the Argand diagram.
Next we apply the same analysis for 2-dimensional vectors given above to 3-dimensional vectors. The cross product is defined and we give the geometrical interpretation. We can now investigate geometrical properties of lines and planes and find distances by using the two types of vector products.
In (ii) we begin with linear equations in two unknowns. These systems are expressed in an array and then as a matrix of numbers. In so doing the different types of systems, namely consistent and inconsistent, ones are defined. We then apply elementary row transformations to the augmented matrix forms and reduce these to the row echelon form. The rules of rank are then used to give the unique or infinite set of solutions for the consistent case or can tell whether the solution is inconsistent.
Next we proceed to operations of addition and multiplication involving matrices. Properties of transpose are discussed. We define diagonal and triangular matrices which results after elementary row transformations are performed. The multiplicative inverse of a matrix whenever it exists is found by using Gaussian Elimination.
A recursive definition of the determinant of a square matrix is given. In so doing the determinant is found and various properties of the determinant are listed that arise out of the recursive definition. We examine the changes made to a determinant by performing elementary row operations. Determinants are used to find the cofactor matrix and then the adjoint matrix from which the inverse is obtained. We finally use the determinant to solve a system of linear equations and explain Cramer’s rule in detail.
Assessment is designed to encourage students to work continuously with the course materials. Active learning will be achieved through assigned problem sheets allowing continuous feedback and guidance on problem solving techniques in tutorials and in the Help-Desk. Assessment will be continual and will be based on the assignments and in-course tests followed by a final examination based on the whole course.
The Help-Desk will be manned by teaching assistants who will give assistance (outside of the formal tutorials) to students on a one-on-one basis or in small groups
Learning Outcomes
Upon successful completion of the course, students will be able to:
- Define a vector and perform basic vector operations (addition, scalar multiplication, length of a vector).
- Find the dot product and cross-product of two vectors.
- Solve equations involving complex numbers and give the solutions in the Argand diagram.
- Give a geometrical proof of triangle inequality.
- Use de Moivre’s theorem to find roots of complex numbers.
- Give geometrical applications of the dot product of 3-dimensional vectors.
- Find the symmetric and parametric equations of a line.
- Find distance between skew lines.
- Find the equation of a plane and the distance between a point and a plane.
- Perform basic matrix operations.
- Use Gaussian Elimination to solve systems of linear equations.
- Find the determinant of square matrices.
- Find the cofactor matrix and use it to find the inverse of a matrix.
- Solve a system of equations by making use of determinant.
How To Study For This Course
“There is no royal road to Geometry” response by Euclid to King Ptolemy’s request for a quick introduction to geometry.
Mathematics is not a spectator sport. In order to develop understanding and problem solving skills it is essential that students do many exercises and problems, including many problems additional to those set in assignments. This is the best way to learn the material properly and also to revise. One cannot learn Mathematics by simply reading the notes given in class and looking over worked examples.
Mathematics and Actuarial Science majors will need to know and understand this material thoroughly and will use it in future courses. It must become firmly entrenched in long-term memory, not something to be forgotten the day after the final examination. Again, the best way to do this is through a great deal of practice in solving problems. The Schaum Series book is a very valuable source of problems and exercises in Linear Algebra.
In addition to your lectures, there is a compulsory tutorial. It is very important that you attend and participate. Do not go just to ’get the notes’. The purpose of the tutorial is to assist you in understanding what was done in the lecture. The Department also runs a Help Desk. There you may get one-on-one assistance from Tutors. Check the Department or MyeLearning for details.
Many students buy textbooks but never use them. They try to rely only on lecture notes. Read your Textbook! Do not be solely dependent on your lecturer. Textbooks give you a second perspective on the material covered in the lecture. They are also valuable sources of additional problems and exercises. In particular, the Schaum Series book on Linear Algebra is a very valuable source of problems and exercises in Linear Algebra.
Course Content
Vectors in the Euclidean Plane:
- Algebraic definition of a vector .Geometric interpretation of a vector. The norm of a vector. Unit vectors. The Triangle Inequality.
- The dot product. Projections and geometric applications of the dot product. Parallel and perpendicular vectors.
Complex Numbers:
- The vector interpretation of a complex number.
- Curves and regions in the Argand Diagram.
- Geometric proof of the Triangle Inequality.
- Roots of polynomials and de Moivre’s theorem.
Vectors in 3-dimensional Euclidean Space:
- Unit vectors and direction cosines. The norm of a vector.
- The dot product and its properties. Projections and geometric applications of the dot product. Orthogonal vectors.
- The cross product and its properties. Geometric interpretation of the cross product.
- The symmetric and parametric equations of a line. Intersecting, parallel and skew lines. Shortest distance between skew lines.
- The equation of a plane. Intersecting and parallel planes. Distance between parallel planes.
Systems of Linear Equations:
- Linear equations in two unknowns. The general case of m linear equations in n unknowns.
- Introduction of a matrix as an array of numbers. Gaussian Elimination and augmented matrices. Row echelon form and reduced row echelon form. Consistent, inconsistent and over determined systems.
Matrix Algebra:
- Equality of matrices. Addition, scalar and matrix multiplication. Square matrices and non-singular matrices.
- The transpose of a matrix. Diagonal and triangular matrices. Finding the inverse of a matrix using Gaussian Elimination.
Determinants:
- Recursive definition of the determinant. Evaluation of determinants. Properties of the determinant.
- Effect of elementary row (and column) operations on the determinant of a matrix. Determinant of a triangular matrix.
- The adjoint matrix. The inverse of a matrix in terms of its determinant and its adjoint matrix.
- The solution of a system of linear equations using the determinant. Cramer’s Rule.
Teaching Methodology
Lectures: Two (2) lectures each week (50 minutes each).
The primary mode of instruction is the face-to-face lecture. Lecturers vary in how they present the material but most will use additional modes (multi-media) of presentation in their lectures.
Tutorials: One (1) tutorial session for eleven weeks (50 minutes each) – These tutorials will serve as a forum for discussing concrete examples that students will encounter in their respective fields. Difficulties in the assignments will be discussed there. You may also be asked to solve problems during class, to work in groups to solve problems and to do small problems, and to make small presentations.
MyeLearning
MyeLearning is a valuable site for all students and is the main communication tool between students and lecturer. Most of the course materials and information will be found there.
Assessment
Final Examination (one 2-hour written paper) - 50%
Coursework: 50%, consisting of two, equally weighted in-course tests (40%) and assignments (10 %).
You will be given one assignment per week (a total of 10), except when coursework exams are due, all of which will be graded. Nine assignments will count towards the total mark.
The final examination consists of essay type questions involving definitions, statements of theorems, proofs of major theorems and problems. The coursework examination consists of shorter essay type questions which focus more on knowledge and understanding and solving simpler problems. The final examination will place greater emphasis on problem solving skills. The assignments consist of problems based on lectures that are covered in each week. These questions are compulsory. They are given out each week, as indicated in the course calendar, whenever there is no coursework examination.
Course Calendar
Week |
Lecture subjects |
Tutorials/Exams |
---|---|---|
1 |
Introduction/Course Overview Vectors in the Euclidean Plane: Algebraic definition of a vector. Geometric interpretation of a vector. The norm of a vector. Unit vectors. The Triangle Inequality. |
Assignment 1 Given |
2 |
Vectors in the Euclidean Plane: The dot product. Projections and geometric applications of the dot product. Parallel and perpendicular vectors. |
Assignment 2 Given. Assignment 1 due. |
3 |
Complex Numbers: The vector interpretation of a complex number. Curves and regions in the Argand Diagram. |
Assignment 3 Given. Assignment 2 due. |
4 |
Complex Numbers: Geometric proof of the Triangle Inequality. Roots of polynomials and de Moivre’s theorem. |
Assignment 4 Given. Assignment 3 due. |
5 |
Vectors in 3 – dimensional Euclidean Space: Unit vectors and direction cosines. The norm of a vector. The dot product and its properties. Projections and geometric applications of the dot product. Orthogonal vectors. The cross product and its properties. Geometric interpretation of the cross product. |
Assignment 5 Given. Assignment 4 due. |
6 |
Vectors in 3 – dimensional Euclidean Space: The symmetric and parametric equations of a line. Intersecting, parallel and skew lines. Shortest distance between skew lines. The equation of a plane. Intersecting and parallel planes. Distance between parallel planes. |
Coursework Exam #1: (15%) |
7 |
Systems of Linear Equations: Linear equations in two unknowns. Examples of consistent, inconsistent and over determined systems. The general case of m linear equations in n unknowns. Introduction of a matrix as an array of numbers. |
Assignment 6 Given. Assignment 5 due. |
8 |
Systems of Linear Equations: Gaussian Elimination and augmented matrices. Row echelon form and reduced row echelon form. consistent, inconsistent and over determined systems. |
Assignment 7 Given. Assignment 6 due. |
9 |
Matrix Algebra: Equality of matrices. Addition, scalar and matrix multiplication. Square matrices and non – singular matrices. The transpose of a matrix. |
Assignment 8 Given. Assignment 7 due. |
10 |
Matrix Algebra: Diagonal and triangular matrices. Finding the inverse of a matrix using Gaussian Elimination. |
Assignment 9 Given. Assignment 8 due. |
11 |
Determinants: Recursive definition of the determinant. Evaluation of determinants. Properties of the determinant. Effect of elementary row (and column) operations on the determinant of a matrix. |
Coursework Exam #2: (15%) |
12 |
Determinants: Determinant of a triangular matrix. The adjoint matrix. The inverse of a matrix in terms of its determinant and its adjoint matrix. The solution of a system of linear equations using the determinant. Cramer’s rule. |
Assignment 10 Given. Assignment 9 due. |
13 |
Revision |
Assignment 10 returned. |
Required Reading
Essential Text
- S.Grossman. Elementary Linear Algebra. Brooks/Cole, 5^{th} edition, 1994.
- S.Leon. Linear Algebra with Applications. Pearson, 8^{th} edition, 2008.
- P.R.Halmos , Linear Algebra Problem Book . Dolciani Mathematical Expositions. The Mathematical Association of America, 1996.