Tag Archives: M115AH

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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:

  1. If \(E\) is idempotent (\(E^2 = E\)) and normal (\(E^* E = E E^*\)), then \(E\) is self-adjoint (\(E = E^*\)).
  2. If \(E\) satisfies \(\|E v\| \leq \|v\|\) for all \(v \in V\), then \(E\) is an orthogonal projection onto its image.
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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 property of the natural numbers. Like induction, the well-ordering principle is frequently used to prove properties which hold for every natural number. The following is a common strategy for applying the well-ordering principle to prove that every natural number satisfies some property \(P\):

  1. Suppose \(S\) is the set of natural numbers that do not possess property \(P\).
  2. By the well-ordering principle, if \(S\) is not empty it must contain a smallest element, \(x_0\).
  3. Construct some \(x < x_0\) which does not satisfy \(P\), hence is also contained in \(S\).
  4. The existence of \(x\) contradicts the minimality of \(x_0\), implying that \(S\) has no minimal element.
  5. Therefore, \(S\) must be empty (by the well-ordering principle), hence \(P\) holds for every natural number.

This proof technique differs from mathematical induction in that the argument uses contraposition (modus tollens) rather than direct proof (modus ponens).

We are now ready to apply the technique described above to prove the main result.

Theorem Let \(V\) be a vector space over \(\mathbf{F}\) and \(T : V \to V\) a linear operator. Suppose \(v_1, v_2, \ldots, v_m\) are eigenvectors with corresponding eigenvalues \(\lambda_1, \lambda_2, \ldots, \lambda_m\) where \(\lambda_i \neq \lambda_j\) for all \(i \neq j\). Then the set
\[
S = \{v_1, v_2, \ldots, v_m\}
\]
is linearly independent.

Proof Suppose to the contrary that \(S\) is linearly dependent. That is, there exist \(a_1, a_2, \ldots, a_m \in \mathbf{F}\), not all zero, such that
\[
\sum_{i = 1}^m a_i v_i = 0.
\]
Let \(k\) be the smallest natural number such that there exist \(k\) indices \(I = \{i_1, i_2, \ldots, i_k\}\) and coefficients \(a_{i_1}, a_{i_2}, \ldots, a_{i_k} \neq 0\) satisfying
\[
\sum_{j = 1}^k a_{i_j} v_{i_j} = 0.
\]
Such a minimal \(k\) exists by the well-ordering principle. Since the \(v_i\) correspond to distinct eigenvalues, we have
\[
(T – \lambda_i I) v_i = 0
\]
while
\[
(T – \lambda_i I) v_j \neq 0 \text{ for } i \neq j.
\]
Therefore,
\[
(T – \lambda_{i_1} I) \sum_{j = 1}^k a_{i_j} v_{i_j} = \sum_{j = 2}^k b_{i_j} v_{i_j} = 0
\]
where \(b_{i_j} = a_{i_j}(\lambda_{i_1} – \lambda_{i_j})\). In particular, \(I’ = \{i_2, i_3, \ldots, i_k\}\) is a set of \(k-1\) nonzero coefficients whose corresponding linear combination of the \(v_i\)s is zero. This contradicts the minimality of \(k\). Therefore, no nontrivial linear combination of the vectors in \(S\) can be zero, so \(S\) is linearly independent.∎

teaching

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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Basic Logic for Linear Algebra

This Tuesday in class, we will talk about basic logic and the field axioms. Since these topics aren’t explicitly covered in the text for the class, I wrote up some supplemental notes. You can download these notes here. As usual, let me know in the comments below if anything is incorrect or unclear. The “exercises” in the essay are just for your own edification — you will certainly not be asked to submit solutions!

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Welcome to Math 115AH

This post is just an announcement welcoming my students to Math 115AH (linear algebra) this quarter. I am super excited to be working on this course! I am tentatively planning on holding office hours for the course on Tuesday and Thursday after class (12 – 1) in my office, MS 3915B.

You can find Professor Gieseker’s course website here.

I will occasionally post material relevant to class on my blog. Entries (such as this one) will be tagged M115AH.