What is a determinant?

Inspired by this Reddit thread which was inspired by this MathOverflow thread. Prerequisites: matrix multiplication. The original version of the post was missing the minus sign in the definition of the determinant; thanks to my friend Josh Zelinsky for politely pointing that out.

A determinant is the real number \det A associated to a n \times n matrix A = (a_{i,j}), in other words

A = \left[ \begin{array}{cccc} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n,1} & a_{n,2} & \cdots & a_{n,n} \end{array} \right],

such that

\det A = \displaystyle \sum_{\sigma \in S^n} (-1)^{\text{sgn}(\sigma)} a_{1,\sigma(1)} a_{2,\sigma(2)} \cdots a_{n, \sigma(n)}.


This definition, while correct and more concise than the expansion-by-cofactors algorithm used to compute it (which I gave as the definition in MTH 237 because like fun am I introducing permutations to use once in a month-long course), is completely nonilluminating. In English, the definition says that you multiply together every entry in a path across the matrix such that in each path you never repeat rows or columns and add the results. (One such product would be a_{1,2}a_{3,4}a_{2,3}a_{4,1}; notice each row and column is used only once.) This still tells us nothing about what a determinant means or why one gives a shit, and since determinants are one of the most useful tools by which to understand linear operators / square matrices, it’s a pretty big gap to ignore.

It is necessary to add an interpretation to our (definition, algorithm) pair.

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Mathematics is the language of reality, and so eight or nine times a day I find myself casually using a word I learned in topology or linear algebra in a completely different setting. My friends are threatening not to speak to me anymore. But these are good words! If only I had a way to explain the words I’ve come to love without asking my friends to take a math major first—oh right. I have a blog. (The post proper is aimed at an audience with maybe high school math. The footnotes are for mathematicians or an interested reader who wants to know more. The exercises are for anyone with the listed prerequisite.)

Two things are orthogonal if they are perpendicular. But why don’t I just use the word “perpendicular” with retail customers and at the bar? Why roll to lose an audience with a slightly harder word that means effectively the same thing?

Shade of meaning is everything, but where perpendicular lines are going in the opposite direction on a piece of paper, orthogonal objects are going in the opposite direction on everything. There is no setting in which these two objects are connected, correlated, or drift-compatible.

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The tao of calculus

The title of the last article was almost “Who needs calculus?” Then I remembered my personal moral code—no clickbait titles, not even in the face of Armageddon—and changed it to “Calculus for everyone.” Calculus for everyone is, additionally, the imaginary title of the imaginary textbook for the imaginary course described in that article. There are two parts to the book—a How and a Why of calculus—so I thought, why not a Tao of calculus?

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Calculus for everybody

It’s important to make sacrifices for your craft. Today, for instance, I had to pony up $12 to the Wall Street Journal so that I could read about why calculus is so last century. (You will have to do the same.) The article’s title (I confess that at the writing of this opening paragraph I haven’t read past the paywall yet) reminded me that I have a whole soapbox about why everyone and their grandmother seems to take calculus, despite the large swath of its content being useless to all but engineers, physicists, math majors, and the occasional business or biology major. Let’s get started!

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Trigonometry teachers HATE HIM

One weird trick! Click here!

Somewhere in my years of “tutoring calculus,” also sometimes referred to as “tutoring trigonometry,” I discovered a trick (here meaning a tool devoid of any conceptual knowledge of the topic at all) for evaluating the cosine (and therefore also sine) of the upper-right quadrant special angles 0, \pi/6, \pi/4, \pi/3, \pi/2. This can be extended to the rest of the unit circle in the usual way.

I rarely see this move discussed, and it is such an easy way to “memorize the unit circle” that I find this surprising. As with all academics, I should be writing, so here you go.

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