**Subject Code/Name:** MTH2021 - Linear Algebra with Applications **Workload:** 3 x 1 hour lectures per week

1 x 2 hour tutorial per week (

**starts Week 2, not strictly compulsory, but you'll be attending a few weeks anyway)****Assessment:** Three written assignments (6%, 7%, 7% respectively)

One midsemester test (10%)

Final exam (70%)

**Recorded Lectures:** Yes, with screen capture.

**Past exams available:** Yes, two available. One has answers.

**Textbook Recommendation:** H. Anton, C. Rorres,

*Elementary linear algebra (applications version)* (10th ed. in Sem 1 2013). Not a compulsory buy. Did not consult much throughout semester.

**Lecturer(s):**Weeks 1-6: Tim Garoni

Weeks 7-12: Jerome Droniou

**Year & Semester of completion:** Semester 1, 2013

**Rating:** 4.5/5

**Your Mark/Grade:** Unknown at this point.

**Comments: ** This unit basically extends upon the concepts covered in part of MTH1030 (the matrices, Gaussian elimination, eigenvalues/eigenvectors part), and also tries to generalise some of the concepts of vectors into a more abstract sense.

The ordering of topics here doesn't quite reflect the order in which they will be covered in semester, but rather, a grouping that reflects the links between the familiar and the more abstract.

Interpretation of linear systems, Gaussian/Gauss-Jordan elimination, elementary row operations, matrix operations, determinants and inverses (largely a revision of MTH1030 bringing everyone up to speed, this will be at the start of semester).

General vector spaces - introduces the idea that there are really a LOT of things that can be called 'vectors', goes through some of the properties of these, including span, linear independence, basis and change of basis, dimension, subspaces. Links back to matrices are found in row space, column space, rank, nullity. This is really the foundation of a lot of the unit, so it's good if you understand the concepts here.

Dot products, angles between vectors, scalar and vector projections, magnitude and distance between two vectors are then covered, which is also revision from Spesh/MTH1020/MTH1030. There are a couple of new things (like the matrix of orthogonal projection), but most of the stuff is revision. Later on in the semester, these concepts are learnt in a more abstract sense. Instead of dot products, you now have inner products and inner product spaces. Instead of perpendicular vectors, you now have orthogonal vectors. Instead of magnitude, you now have the norm of a vector.

Matrix transformations/linear transformations - using matrices to transform vectors into other vectors (think rotations, reflections, stretches and skews, as well as just turning vectors into other vectors). Later on in the semester, we get the more abstract 'general linear transformations'. Now, we're no longer just mapping vectors from R^m to R^n, but from any vector space to any other vector space. Isomorphism, onto and one-to-one transformations, linearity of transformations and change of basis are covered here.

Eigenvalues and eigenvectors, eigenspaces, similarity and diagonalisation, as well as applications to quadratic forms, internet search engines, multivariable calculus, solving systems of differential equations. Later on, this gets combined with inner products/orthogonality to form 'orthogonal diagonalisation' (one of the best things Jerome will ever say with his French accent).

Finally, the last chapter is about some other applications in probability (if you remember Markov chains from Year 12 Methods probability, and how we used matrices there, it's a bit like that), and a bit about coding, and a bit about fields and modular arithmetic.

On the whole, this was also a pretty good unit for me (compared to MTH2010, for example). Both Tim and Jerome put in a decent effort to explain things, as opposed to just filling in the lecture notes booklet. On that note,

**I highly recommend you get the lecture notes booklet, and fill it in as we go through the course.** Basically, it will have the theory already there, and the examples/proofs will be filled in by Tim and Jerome as you go through the course. I just found it to be a nice, organised way of having all the lecture notes with you at one time.

The tutorials were OK, you collect a particular week's problem sheet, and bring it to the tutorial where the demonstrator is there to help with any questions/do some examples on the board, etc. I don't think they're strictly compulsory, but you have to attend the tutorials sometimes anyway (in order to hand in assignments and sit the midsemester test), so you may as well stay

.

I honestly didn't use the textbook at all, except for a bit of leisurely reading before the start of semester. It's definitely not required unless you want to learn more or consolidate your knowledge; the lecture notes booklet is what will be used.

In terms of assessment, the assignments are quite doable if you have your lecture notes in front of you and you are capable of following the steps/working out involved. There's usually one or two slightly more difficult questions on each assignment, just to provide a bit of challenge. For one of the assignments, you are encouraged to use a computer to help with your results, so if you can get Mathematica or MATLAB or something of that sort, it might make your life a little easier. (The use of computer software is not compulsory, however).

I found the mid-semester test to be quite doable, as long as you

**understand the concepts.** University maths (from the science faculty at least) is really about

**understanding the concepts.** Some of the more abstract stuff might make your head spin at first (it honestly did that to a lot of people, myself included), but if you can try to have it make sense in your mind, it really makes your life a lot easier. Apparently the median mark for the midsemester test was a fail, so

**please, please, try to understand the concepts.**The final exam was a little easier than I had expected. As Tim Garoni said in our final lecture, "If you want a HD, you

**will** have to be able to do proofs/'show that' type questions. If you just want a C or a D, you can probably get away with just trying the numerical computation style questions." Basically, what he was trying to say is that proofs/'show that' questions aren't a major part of the exam, but will make a difference if you want a good mark. The proofs are usually the easier and simpler-type ones in the lecture, so don't feel as if you have to memorise the incredible number of proofs in the notes. As revision, he suggested going over the mid-semester test (there were similarly-styled questions on the final exam and the mid-sem test). Also, know the key ideas of each chapter well. The vast majority of marks are always for the working out, rather than the answer (I found out to my horror that even in 1-mark questions, you won't get the mark if your working is wrong).

In summary, this unit isn't really all about computations. There is a fair amount of abstract stuff to get your head around, and lots of content is linked, so if you miss a few lectures, it's really easy to have no idea what on earth is going on. The 'applications' aren't the major part of the unit; the theory is. (of course, applications can still be examinable). The content can honestly seem fairly 'dry' and be a seemingly endless maze of "Theory, Proof. Theory, Proof." most of the time. If you like fairly abstract stuff, I think this unit can be a good one to try. However, if you don't, you may find yourself resenting the theory and the way in which it is presented.