Some FAQs (updated now and then).
I’ve excluded the Chapters 17 to 20 from David Lerner’s notes. Does that mean that orthogonal projections, inner product and trace will not be tested?
No. I excluded them because in the notes, more “advanced” concepts are discussed in the chapters. For example, the chapter on inner product introduce the concept of positive definite and bilinear, which may be confusing. However, you will still need to know that how to compute the inner product between two vectors and what it means for two vectors to be orthogonal.
Similarly, orthogonal projection and trace are classic examples of linear transformations. Hence, you should be able to determine the kernel and range of these transformations. For example, Tut 10 Q2 asks you to determine the range of a trace map, which is within your ability. However, we will not ask you to show that trace map is invariant with respect to a change of basis.
I do not understand the concept of determining a matrix associated with a linear transformation.
Please take a look at this video from khan academy. I hope this is more illuminative.
Tut 11, Q4 is wrong.
My apologies. I’ve updated the worksheet.
When am I going to provide the feedback for the Final Sample?
Now. Here are the solutions with some feedback. Do not be too bothered by question 5.
If you want to find out more about projections as linear transformation, you can look at this video.
Am I going to provide the answers for the tutorial fill-in the blanks?
Here are some general guidance to studying linear algebra:
David Lerner’s notes are useful for self-study. They contain small exercises that help understanding and sometimes give hints to the tutorial questions.
Chapters to take note are: Ch 1- 8, Ch 10-13, Ch 15.
Khan Academy also has a comprehensive course for linear algebra too.
Relevant videos/lessons are:
Vector and spaces – Vectors, Linear combinations and spans, Linear dependence and independence, Subspaces and the basis for a subspace, Vector dot (without cross) product, Matrices for solving systems by elimination, Null space and column space.
Matrix transformations – Functions and linear transformations, Linear transformation examples, Transformations and matrix multiplication (matrix products are more relevant), Finding inverses and determinants.
It looks a lot, but the videos are typically very short. You can also select the topics to revise.
For those who prefer the “abstract” flavour, you can search for the text Linear Algebra Done Right by Sheldon Axler. (The link is the link to the official website. The book is easily found online, but I would not place it here.) This text is probably more useful for Linear Algebra 2.
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Good luck for your examinations!