Course Review: MATH 223

Linear Algebra

“This is really useful. You might get a call one day from Downtown, asking you how to take the inverse of a two-by-two matrix.”

Text: Linear Algebra and its Applications by David C. Lay (optional)

Prof: Dr Richard Anstee

Professor Anstee looks kinda like a classic Prof. Fuzzy white hair, check. Kindly yet contemplative expression, check. He is also a natural teacher, having a cavalier attitude yet still able to get the material across. He says the most whimsical things in the middle of lecture, making his classes really entertaining. He spent the first week teaching many of the major concepts intuitively for two-by-two matrices. The following weeks till the second midterm were both applying and more rigorously fleshing out the ideas discussed in the first week, so don’t get psyched by the first week if the material is really new for you. Anstee tends to switch perspectives a lot during lectures, without explicitly informing you. I have a feeling he does this on purpose, to keep us on our toes, because viewing things from multiple perspectives is a core  feature of Linear Algebra. He is also a really kind Prof, holding many-many office hours and always willing to chat to his students. He also brought home-made cookies on the last class :D.

Some quips:

“You never know who you will meet on the bus!”


Difficulty

Since I had learnt this material before, I am not a fair judge. However, for people who have either never taken an honours math class or have no familiarity with matrices this course is probably both challenging and do-able. More then any other course I have taken so far, conceptual understanding is key. In calculus you can memorize a ton of algorithms and you can go pretty far. In linear, the questions are deceptively simple if you can make the connection to theory. On exams, some ten mark questions were literally two or so lines. He also gives you practice exams where the first few questions (these are often the ones that test computation) are exactly the same as the one on the real exam, except with different numbers, so there is really no excuse for losing many marks on the first 40%. Some of the homework questions are significantly more challenging then exam questions, but they are do-able, and he gives you lots of help if you ask him. Conceptually, since linear algebra is about multiple perspectives, to do very well in this course, I would suggest consulting a variety of sources. In terms of marks, after generous scaling, I think that one can get a decent grade in this course, though those who are new to honours math might have to work extra hard.


Key Concepts

Representing linear transformations as matrices

N-dimensional Vector Spaces (generalization of R^n)

Properties of matrices/transformations (characteristic, determinant, trace, rank, eigenvalues)

Types of matrices/transformations (diagonal, diagonalizable, invertible, orthogonal, hermitian)


 Hard Concepts

Change of basis: Got really confused by this. What I realized was, depending on the context, one can interpret a matrix as a change-of-basis, or as a linear transformation on a given basis. Another thing to keep in mind is that a change-of-basis matrix has column vectors that form a basis of the domain in the co-ordinates of the image.

Complex conjugate: Didn’t go to deep into complex numbers, but got confused about which theorems apply in the complex/real case respectively.

Entry-wise manipulation of arbitrary matrices: Had difficulty visualizing the a_ij element in a matrix.


Conclusion

Fun, intelligent class. Favourite class this term.

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