Elementary Linear Algebra Autumn 2008
Solving systems of equations is an important and difficult
mathematics. Even figuring out if a single equation has any nontrivial solutions (e.g. Fermat’s
Last Theorem) can be extremely difficult. If the system consists of linear equations, there is
an algorithm to find it’s solutions. In this course, we will study this algorithm, which will
help us solve many problems in both algebra and geometry.
The techniques we develop apply not only to solutions of
linear systems, but also to many
different mathematical systems, e.g. complex numbers, real valued functions, lines through
the origin in the plane, etc. When we abstract the properties that allow us to apply our
techniques to these systems, we get what is called a vector space. The benefit of working with
vector spaces is that everything we deduce about them automatically applies to all the diverse
systems we want to understand. In the latter part of the class we study vector spaces.
Course Objectives: The successful student will demonstrate:
1. Ability to translate between systems of linear
equations, vector equations, and matrix
equations, and perform elementary row operations to reduce the matrix to standard forms.
2. Understanding of linear combination and span.
3. Determination of the existence and uniqueness of a system of linear equations in terms
of the columns and rows of its matrix.
4. Ability to represent the solution set of a system of linear equations in parametric vector
form and understand the geometry of the solution set.
5. Understanding of linear dependence and independence of sets of vectors.
6. Understanding of linear transformations defined algebraically and geometrically, and
ability to find the standard matrix of a linear transformation.
7. Understanding and computation of the inverse and transpose of a matrix.
8. Understanding and computation of the determinant of a matrix and its connection with
9. Understanding of the notions of a vector space and its subspaces and knowledge of their
10. Knowledge of the definitions of a basis for and the dimension of a vector space, and
ability to compute coordinates in terms of a given basis and to find the change of basis
transformation between two given bases.
11. Ability to find bases for the row, column, and null spaces of a matrix, find their dimensions,
and knowledge of the Rank Theorem.
12. Ability to find eigenvalues and eigenvectors of a matrix.
13. Knowledge of all aspects of the Invertible Matrix Theorem.
14. Knowledge of the Diagonalization Theorem and ability to diagonalize a matrix.
The achievement of these goals will be measured by three exams and a final.
Homework: For each section of the text, I have
posted problems at the URL:
Most problems are odd numbered so that you can check your answer in the book. After
a section is presented in class, you should attempt to solve the corresponding problems.
Homework is a major part of the learning process in Mathematics and it is absolutely essential
to your success that you do the homework.
The homework will not be collected!
I will set aside time Thursday and as needed
to go over homework problems that you need
Exams: There will be three fifty
minute exams and a final: the first exam is on Friday, Oct.
10, the second exam is on Tuesday, Oct. 28, and the third exam is on Tuesday, November 18.
The final exam is on Friday, Dec. 12, 8:00-10:00 am.
Grading: Your grade will be based on the exams and the final, as follows:
• exams: 65 %
• final: 35 %
Makeup Policy: There will be no
make-up, early or late quizzes or exams. If some health
or family emergency prevents you from taking an exam, you should contact me immediately
before the exam. If you must miss an exam due to an emergency, see me and we can make