# Two Tricky Linear Algebra Problems

There are a couple questions from the homework that seem to have given people (myself included) a fair amount of trouble. Since I wasn’t able to give satisfactory answers to these questions in office hours, I thought I’d write up clean solutions to the problems, available here. The questions both involve projections $$E : V \to V$$ where $$V$$ is an inner product space. The problems ask you to prove: If $$E$$ is idempotent ($$E^2 = E$$) and normal ($$E^{\ast} E = E E^{\ast}$$), then $$E$$ is self-adjoint…

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# Math 115AH Midterm 1 Review Questions

I’ve written up a few review questions for the first midterm for Math 115AH, available here. I will go over solutions to any of the problems during class or the review session (which will be Thursday, April 30 from 5 to 7 PM in Boelter 2444). Let me know in the comments below if you notice any typos or if you have questions about any of the problems.…

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# Logic and Sets

I have just uploaded notes on Basic Logic and Naive Set Theory for math 115AH. Please let me know in the comments below if you notice typos or if anything is unclear.…

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# Independence of Eigenvectors and the Well Ordering Principle

In linear algebra, we frequently use the fact that a set of eigenvectors with pairwise distinct eigenvalues is linearly independent. Hoffman and Kunze prove this fact (see the proof of the second lemma on page 186) by using some elementary facts about polynomials. Here we prove the linear independence of eigenvectors using the well-ordering principle: Well-ordering principle Suppose $$S$$ is a non-empty subset of the natural numbers, $$S \subseteq \mathbf{N}$$. Then $$S$$ has a minimal element. The well-ordering principle is equivalent to mathematical induction, and is thus a fundamental…

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# Naive Set Theory

I have posted some notes on naive set theory, available here. They cover the basic algebra of sets and functions. Many of the important identities are left as exercises to reader. A bonus section on Russell’s paradox shows that this “naive” approach to set theory is not sufficient for a rigorous theory of sets–axiomatic set theory is the only way to go.…

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