Linear algebra book alpha release

ivan

The No bullshit guide to linear algebra is almost finished. I know I have been saying something along these lines for almost two years now, but it’s for real this time. Read below for a general preview of the new chapters, and the story about why it took so long to write them.

I was underestimating the creative effort that was needed to “close” the book on a good note. Students having survived through seven chapters of theory (vectors, linear transformations, matrix representations, vector spaces, etc.) deserve to hear about some interesting applications of LA. Makes sense. This was my initial idea around Nov 2014. It took 1.5 years to write the applications chapter because linear algebra has a lot of applications!

I started from a standard list of LA applications, but I wanted to talk about several other topics like probability theory and quantum mechanics. In the end, the book will have three applications chapters, the main of them being:

  • Chapter 8: Applications
    • Balancing chemical equations: this is a standard procedure for finding the stoichiometric coefficients in chemical equations.
    • Input–output models in economics: suppose you have a formula that defines a vector $\vec{v}$, but the coefficient $v_1$ depends on the other coefficients $v_2,v_3,\ldots$. How can you find $\vec{v}$ from this self-referential formula? If the dependence between coefficients is linear, we can express the formula as a matrix equation and solve it using standard linear algebra techniques.
    • Electric circuits: I did my undergrad in electrical engineering, so for sure I’ll have to discuss solving the systems equations that result what you apply Kirchhoff’s circuit laws in a circuit consisting of batteries and resistors.
    • Graphs: only the basic notion of an adjacency matrix is introduced.
    • Fibonacci sequence: an cool eigendecomposition trick for computing the powers of the matrix $A=\begin{bmatrix} 1 & 1 \\ 1 & 0\end{bmatrix}$, and obtaining a fast way to compute the any term in the Fibonacci sequence.
    • Linear programming is a topic that has nothing to do with linear algebra per se, but also uses. In the end I decided to cut this section from the book, and offer it as a free tutorial on github. It’s worth reading if you’re interested in this stuff, but there’s no need to make everyone read it. Though this tutorial could be a good starting point to discuss other optimization problems, quadratic programming, KKT conditions, etc.
    • Least squares approximate solutions: reuses some material from a past HN comment and adds some more detail and an example least squares fit (thx Plot.ly!).
    • Computer graphics. Writing this section took me about one month. The main slowdown was the fact that I didn’t know anything about computer graphics. After reading a few books I found online, wikipedia, and some online tutorials about 3D programming (WebGL, OpenGL), I was able to put together a compact introduction to homogeneous coordinates and their applications to computer graphics.
    • Cryptography and Error correcting codes. These topics are further from the “main idea” of linear algebra, but I thought it would be cool to show some of the more useful applications of using finite fields $F_q = \{0,1,2,\ldots,q-1\}$.
    • Fourier analysis is the bread-and-butter skill for understanding signal processing. Since the Fourier transform is analogous to a change of basis operation (from the time basis to the frequency basis), I figured I can cover this too. In particular I found a nice motivating example that introduces the “family of orthogonal functions.”

I thought about stopping here, but I really wanted to say something about probability theory and quantum mechanics. At the same time, I don’t want to make the book too long either, so I also thought about cutting the probability and QM sections. Readers advised me to keep them, and in the end two whole chapters were added:

  • Chapter 9: Probability theory
    • Probability distributions: really high level description of probability distributions $p_X(x) \equiv \textrm{Pr}(\{X=x\})$, and an introduction to conditional probability distributions.
    • Markov chains: mini intro on essential material on Markov chains and stationary distribution.
    • Google’s PageRank algorithm: a worked example of the PageRank Markov chain for a small graph of “webpages” that shows the general idea.

With the basics of probability covered, and a whole book of linear algebra, I was finally in a position to cover some topics in quantum mechanics.

  • Chapter 10: Quantum mechanics
    • Polarizing lenses experiment: describes a simple table-top experiment with light that is analogous to the Stern-Gerlach experiment with electrons. The outcomes of the experiment cannot be understood using “classical” physics paradigm, and thus serve to motivate some aspects of quantum models.
    • Dirac notation for vectors: $|v\rangle$ is nicer than the usual vector notation $\vec{v}$, specially when computing complex conjugates, and inner products.
    • Quantum information processing: yet another intro chapter when quantum processing ideas are introduced through an analogy with digital systems. Analog-to-digital converters are like quantum state preparation, and digital-to-analog converters are like quantum measurements. It’s a bit of a strenuous analogy, but it reinforces the “system thinking” which I thought would be useful for readers who have never seen quantum before.
    • Postulates of quantum mechanics: this section is at a level appropriate for a graduate-level intro to QM course. No BS here.
    • Polarizing lenses experiment revisited: shows how using quantum models for states (vectors) and measurements (projections), leads to the correct predictions about the polarizing lenses experiment.
    • Quantum physics is not that weird: this is another “fluff” section, where I demolish various misconceptions about QM. It’s not that weird, it’s just vectors…
    • Applications: of QM cover lots of topics, like particle physics, solid state physics, superconductors, quantum cryptography, and quantum computing.

Overall I’m quite happy with the way Chapter 10 turned out, but I’m worried there are too many words. Since I know most people will be afraid of the math, I wanted to explain QM in words as much as possible, but I might have gone overboard. Maybe the chapter could have been shorter/simpler if I just hit the reader with the math right from the beginning, and not explain all about math models for reality, and the different areas where quantum mechanics is used? I guess we’ll have to wait and see what the readers think…

I am now doing a final proofreading pass on paper, before shipping the book to the editor so she can work her magic. Once that’s done, I’ll send out an update to all the current readers with the “alpha” release. Any readers who have purchased the PRE-RELEASE version of the LA book should feel free to email me if they want to see the latest draft.