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Review of core concepts

How to review

Let's talk about the purpose of the review, and how to go about it:

Linear algebra often feels like an alien landscape because it's a kind of math we haven't encountered before, unlike calculus which we saw bits and pieces of in high school.

It can feel overwhelming to have so many concepts that build on each other come at you in rapid succession.

What's important to know: you aren't alone and it's perfectly natural to experience confusion, and a lack of confidence in "knowing that you know it".

The purpose of this review is to go over basic ideas again, because the more times you review, the more you will get comfortable with the core ideas (and notation).

The wrong way to do this review is to just read.

What you should do:

Get your notebook out.
For every section: read and then write it out in your notebook, explaining it to yourself.
Add your own examples alongside mine. For example if I make an example vector \((2,3)\), you should make a different one.
Bottom line: if your written notes are smaller than the size of the review, something's wrong.

Important: Do NOT read the review topics and think "Oh, this is all that's really important, I can skip the others". The topics covered here are fundamental and intended to anchor your understanding of the other topics.

Shall we begin?

Do the review

The review is divided into three parts:

Part-I: Vectors, dot-products, lengths, angles, orthogonality, matrix-vector multiplication, solving \({\bf Ax} = {\bf b}\) exactly and approximately.

Part-II: matrix-matrix multiplication, spaces, span, basis, independence, orthogonality (again), projections.

Part-III: summary of useful results, summary of some key theorems.

What else to review

The next steps are to review the following:

Read sections 4.10, 4.11, 4.12 from Module 4.

From Module 5:

Work out the RREF example in 5.2 without looking at it.
Review 5.4
Read the statements of propositions 5.1 to 5.9 to understand what they state.

Review 6.7 from Module 6.

Review 7.7 (projections) from Module 7, and the theorem statements in 7.11.

Review the theorem statements in Module 8 to understand what they state.

This completes the review of basic concepts.

After this:

Go back to some of the other sections where you think you need to review, and write down questions you might have.
Keep in mind that Module 10 is challenging, so don't fret if you struggle with some of it.