: Dr Mark MacDonald, email: m.macdonald@lancaster.ac.uk, office: Fylde B12
Mondays: 9-10am and 2-3pm, Tuesdays: 9am-10am (Weeks 1,3,5,7,9 only)
(2 hours): Tuesdays and Wednesdays, Weeks 2, 4, 6, 8, and 10 only.
: 15% Coursework, 85% Final examination. The coursework, which is primarily formative (meaning its purpose is mainly to assist with learning) will consist of:
Written assignments (7%), which will be due on Thursdays at 2pm in Weeks 1, 3, 5, 7, and 9, in your tutor’s pigeon-hole. They will be marked and returned to you at the next workshop. If your pages are not firmly attached together with a staple, then your tutor might remove marks.
Workshop tests (4%): A short assessed test will be given in each workshop. The questions will be based on the workshop exercises.
Weekly true / false quiz (4%), online, due every Saturday at 2pm.
This module has been designed with the assumption that students will actively engage with all forms of assessment. The purpose of having multiple forms of assessment is to encourage a robust understanding of the topics covered.
The MATH220 Moodle page will include assignments, workshop exercises, all solutions, and these notes. Please bring any mistakes or typos in this material to my attention.
On each assignment, you are expected to give an honest and complete account of who helped you on the exercises (names of classmates, or tutors), and for which questions they helped. You may also state other resources you used, such as books or websites. It is okay to work with others on assignments! But you are not allowed to copy other people’s work and claim it as your own.
The aim of this module is to introduce several new concepts in linear algebra, building on knowledge and techniques that have been acquired in MATH105.
Throughout this module we will assume you are familiar with several concepts and techniques introduced in MATH105, such as matrix multiplication, applying row operations, solving systems of equations using the augmented matrix method, translating between linear transformations and matrices, and finding eigenvalues and eigenspaces of a matrix.
In this module we will introduce the concepts of linear independence and basis of vectors, as well as positive definiteness and inner products. These will give us a renewed understanding of familiar concepts of “length” and “angle”. We revisit linear transformations, and learn how to express them as matrices using non-standard bases. Then we will introduce the well-known spectral theorem, which will let us decompose a real symmetric matrix by finding an orthonormal basis of eigenvectors. Finally, we will learn a way to understand non-diagonalizable matrices, by finding the Jordan normal form.
Almost all of the questions in the summer examination will be either identical to, or variations of the exercises in these notes.
You are expected to attempt the in-text exercises as you progress through each Chapter. The word “exercise” means “Now you have been given the resources to solve this problem. Please use those resources and try to solve it.” Part of the puzzle is figuring out which resources you need. Solutions to any of the exercises labelled “(Bonus)” may be submitted directly to the Lecturer, who may award marks that will be added to your coursework.
There is no single textbook which this module follows, but further reading may be found in the Library’s Linear algebra section, which has the code AQN.
The workshop tests will occur at the end of each workshop (once every two weeks), and will be about 15 minutes in length. You will be given the list of potential workshop questions in advance of the workshop; each week the list will contain about 10 exercises from these notes. During the test, you will not be allowed to look at your notes, or have any other resources in front of you. Students will be encouraged to prepare for the tests in groups, but the tests themselves will be taken as individuals.
Each workshop test is to be marked out of 4, using the following marking scheme:
| 4 | Correct and complete solution, with proper use of notation and terminology |
|---|---|
| 3 | Essentially correct solution, with only minor gaps, errors, or notational mistakes; almost all of the relevant knowledge and/or skills have been demonstrated. |
| 2 | The student has made clear progress towards a correct solution; some relevant knowledge and/or skills have been demonstrated. |
| 1 | A small about of relevant knowledge or skills have been demonstrated. |
| 0 | No relevant knowledge or skills have been demonstrated. |
The test will be marked by your workshop tutor, and returned at the following workshop.
: It is hoped that you will take advantage of the workshops in the following two ways: (1) Use the feedback on your work from your workshop tutor to improve your skills and understanding, and (2) Take the opportunity to ask questions about concepts in the module that are unclear to you. Since the workshops are only every two weeks, it is important that you use the workshop time wisely. Note that the final workshop’s test will be replaced with an assessed group presentation, the details of which will be available on Moodle.