A determinant is the real number associated to a matrix , in other words

such that

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 ; 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.

If this all starts to sound a bit abstract, skip ahead to the examples and come back after.

Consider what multiplying an arbitrary real number by a fixed real number (Greek “lambda”) does. If , it destroys the whole number line, collapsing it into its origin, zero. If , it flips the number line over; positive numbers become negative and negative numbers become positive. Finally, the magnitude of , the number , determines how much the length of the unit interval gets stretched.

Let’s kick this up to a two-dimensional plane, where we’ll stay for the rest of the article. In this plane, we replace our arbitrary number with a vector , where gives the horizontal position and gives the vertical position of the vector. Our fixed transformation is going to be replaced with a square matrix

and rather than number multiplication, our matrix does this to :

You may recognize this as regular old matrix multiplication, but more illustrative for us, it is the action on the plane by the matrix . In other words, goes in, comes out.

Let’s look back at the properties of our number multiplication and see how each comes through in the new two-dimensional setting.

- will either destroy information or it won’t.
- will either flip the space inside-out or it won’t.
- will stretch the unit square whose edges are the vectors and by some factor.

It will turn out that the determinant is a single number that encodes whether our matrix destroys or flips and how much it stretches by, just like the number was able to encode all of this information for regular real-number multiplication. First, a consequence of the definition of the determinant *and *the algorithm used to compute it is that

Using this easy formula, let’s do one example that shows each property.

**e.g. ****1. **Suppose

All of the vectors in the unit square, in gray, are stretched and sheared such that they are now inside the blue parallelogram. (The reader can verify this using matrix multiplication; multiplying by any vector whose horizontal and vertical component are less than one will give you a vector inside the blue parallelogram.) The area of the parallelogram is 5, which is coincidentally the same as

The matrix stretches the plane by a factor of 5, and the matrix’s determinant tells us that.

** e.g. 2. **Suppose

If we think of rightwardness as our “first” direction and upwardness as our “second,” then the regular plane is *oriented* counterclockwise; we travel from to as we travel from 3 o’clock backwards to 12 o’clock. Using matrix multiplication,

and

The matrix switches our directions, so the plane is now oriented clockwise. This reversal of direction is just like the reversal of direction we get on the number line when we multiply by -1! In fact:

If a matrix changes the plane’s orientation, its determinant is negative. Since the matrix didn’t stretch the unit square, just flipped it over, its determinant is just -1.

**e.g. ****3. **Finally, let’s have

This matrix sends every vector in the unit square, shown in gray again, to just vectors inside the red vector. The area of a line is zero, but more importantly, we’ve lost a whole dimension! Like multiplying by zero, the matrix destroys something—not everything, but the whole second dimension in our unit square. This gives us a clue that our determinant will be zero. Well,

So it is.

A *definition *is a mathematically precise statement of the form “all *X* are *Y *and all *Y *are *X*.” It is a statement that fully characterizes an object and tells us exactly what it can and cannot do. Like a card in a game like *Magic *(which was not coincidentally made by a mathematician), this card gets to be combined with other cards to produce additional results. For example, if a determinant of a matrix is zero, that matrix is said to be *singular*, and has no multiplicative inverse. The definition of the determinant is that gobbledygook from the top of the article.

An *algorithm *is a process by which an object may be computed and is not necessarily the same as the definition. We compute determinants typically by breaking the matrix up along a row or column into smaller determinants. Though this “definition” is suitable for an undergraduate course because of its conceptual simplicity (you have to dip your toe into abstract algebra to fully understand the regular definition), it is a little long to write down, and at shortest it requires two sums (one for a row and one for a column) and a proof that the choice of row or column doesn’t change the determinant.

An *interpretation *is not at all mathematically precise and it is not good for computation. Rather, interpretation gives you a mental image of the object you are dealing with, something more palatable to a human brain. If anything, it serves as a bridge to the definition, allowing more reliable information storage.

It’s my opinion that all three are equally important, and that the third doesn’t see nearly enough play.

]]>*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.

Let’s define some words.

In **geometry**, lines are **perpendicular **if they mutually form right angles. The ur-example is a completely horizontal line and a completely vertical line, like shown. Any rotation of this figure, additionally, will also exhibit perpendicular lines.

In **vector calculus**, directions in the plane are pairs of numbers (**vectors**) where gives the horizontal movement and gives its vertical movement. The completely horizontal line is in the direction of the vector , and a purely vertical line follows . Some other line might be in the direction (pictured) which, much like plotting points in high school, tells you to go right five units every time that you go up three. [1]

Determining whether two vectors are perpendicular is as easy as multiplying their horizontal coordinates together and adding that to the product of their vertical coordinates. This operation is called the **inner**** product** (for the plane) and goes like this:

The inner product is zero when two vectors are perpendicular.

Don’t let funny-looking angle brackets and semicolons (really only used to differentiate from the comma used elsewhere) scare you off: multiply the first number with first number () and the second number with second number () and add the results. This may be the easiest thing a calculus student learns to do.

**The take-away here is that the inner product measures how much two objects are pointed in the same direction. If that measure comes up zero, they are orthogonal.**

Developing mathematics—remember, we are building tools to describe the natural world—is about **consistency** and **generality**. When we build a new toy like an inner product, we need to make sure it matches what we already know. Are and perpendicular with this new definition, that perpendicular vectors have an inner product of zero? Let’s find out:

* Exercise. *You were once told that if a line has a slope of , its perpendicular line has a slope of . Prove that the vectors and are perpendicular like I just showed you.

Score! So, our new toy is **consistent**. What does it mean for it to be **general**? Ah, we finally made it to orthogonality.

Vectors don’t need to go in only two directions. It won’t surprise you to learn that a vector representing three-dimensional position in space has two “ground” components—forward and back usually given by , left and right usually denoted —and an up and down component usually called .

How do you tell if there’s a right angle between two three-dimensional lines and ? Same way: their inner product

* Exercise. *Are lines in the directions and perpendicular? How about and ?

It’s better than that. In two or three dimensions, the inner product will tell you not just whether or not two vectors are orthogonal, but it will give you a measure of how parallel they are! For those of you that know trigonometry, is proportional to the cosine of the angle between and . [2] For those of you that don’t, that’s fine: the take-away here is that the inner product measures how much two objects are pointed in the same direction. If that measure comes up zero, they are **orthogonal**.

In **linear algebra **and **functional analysis**, we study spaces that don’t have our usual idea of angles. Sometimes the objects we study aren’t even lines—they may be matrices, polynomials, or other functions. [3] But we still want to know how much two objects mutually dovetail. If you can find a big enough collection of **mutually orthogonal vectors**—in other words, a bunch of vectors for which every two are orthogonal—every object you need to talk about can be expressed in terms of this collection. It’s like coordinates, and everything can be represented by some number taken along each coordinate. Kind of like how we use perpendicular lines on a plane to plot coordinates. How about that?

On these alien worlds with lines that aren’t lines, what’s an angle? Well, an angle is an inner product! At this level of abstraction, we **generalize **the notion of an angle to just match the inner product concept that worked pretty well at the lower levels. Our vectors may not exactly be lists of numbers anymore—especially if we are working in infinitely many dimensions [4] instead of two or three—so the usual multiply-and-add-the-coordinates approach will fail. Other approaches work, but must satisfy certain rules [5], and these inner products still calculate “how much” two of your vectors travel together. And if they have nothing to do with each other? They’re orthogonal.

** Exercise. **[Prerequisite: calculus.] Continuous real-valued functions on the interval are vectors and here,

is a satisfactory inner product [6]. Prove that and are orthogonal. (This is the starting point for the very applicable area of mathematics called **Fourier analysis**, which views functions as combinations of waves.)

* Exercise. *[Prerequisite: complex numbers.] Using the rules in [5], discover an inner product of two complex numbers and determine whether 1 and are orthogonal with scalars coming from the real numbers.

In many settings, both concrete and abstract, there is a tool called an inner product (sometimes a **dot product** in calculus) that measures the “parallelness” of two objects—in more sophisticated language, to what degree these objects are lined up, similar, or compatible. The word “perpendicular” is generally reserved for lines on a plane, or lines in general. But if you *really *want to break up with your boyfriend properly, in infinitely many dimensions, “orthogonal” is the word you’re looking for.

**Footnotes:**

[1] How am I getting away with conflating vectors and lines here? Mostly, it’s because I don’t want to overwhelm new readers with too much detail. A line through the origin is equivalent to all multiples of a single vector. Other lines are pushed out to a certain point, and *then *identified with a span of vectors. Let’s imagine that where the lines we’re talking about meet is the origin and then it should be okay.

[2] What does cosine have to do with whether two lines are parallel? Remember that the cosine of a horizontal rightward angle is 1, but as your angle becomes less horizontal and more vertical, its cosine shrinks and its sine grows. The inner product treats as the horizontal axis and measures the angle between that line and .

[3] Well, they kind of *are*. From a certain point of view, their appearance is linear, and not bendy, which is required to have an inner product at all (see [5]).

[4] As you do.

[5] An inner product with scalars (numbers, usually) coming from a set with division defined must have:

- where is the complex conjugate,
- where $lambda \lambda$ is any scalar, and
- with equality if and only if .

[6] If you were unlucky enough to learn your calculus from me, then you know this is exactly the infinite-dimensional version of multiplying your coordinates and adding.

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What does that even mean? This is how I look at it: Mathematics is the most valuable tool we have for understanding the world around us. Why not, then, use it to understand the world within us as well? Certain mathematical concepts have an almost spiritual quality. I’ve had professors describe the central idea of a topic as its “moral,” where the pun works completely. I refer to grokking an idea, which Wikipedia defines as

to understand intuitively or by empathy, to establish rapport with — to empathize or communicate sympathetically (with); also, to experience enjoyment

*To establish rapport with. *It makes itself sound like we’re making *friends *with the universe. *I have shaken the hand of linear algebra.*

This is the sort of hippy-dippy almost pseudo-mathematics that can make some readers angry; if you are one such reader, feel free to leave. I don’t propose replacing any curriculum with what I’m about to say. It’s just what my mathematical perspective has told me about the way I sometimes feel.

You know, because we’re *friends*.

Calculus, in case you didn’t know or don’t think of it this way, is about change. Specifically, you have **functions**—observable phenomena—dependent on **variables.** The change in a function given some unit of change in a variable is called a **derivative.** Studying derivatives, more or less, is the first semester of calculus.

In many settings, the calculus student is completely unconcerned with where a function sits, in other words, its absolute location. The functions , which squares its input, and , which does the same but then adds three, when graphed on a pair of axes sit above and below one another. However, they have the same derivatives with respect to the input .

So the derivative destroys positional information. Why is this important? Consider a car ride: Your sense of the external tells you *where *you are: I-20 heading out of Mississippi. Your **second derivative** (*i.e. *the derivative’s derivative) is your **acceleration**, and you don’t just see that on the speedometer. You *feel *that. Something is whispering to you from the dark places where mathematics speaks.

Let’s suppose this is something that generalizes. Position is seen; derivatives are felt. Does this idea extend to other examples? I think it does. I have both a personal example and a more important, broader example to share.

I graduated recently and didn’t go straight into a “proper” job in my field. I work as an adjunct instructor now, but I still do some mostly-unrelated part time work. My situation has been steadily improving since August, and now I am *terrified *that I will regress at all. Objectively, that’s stupid. If I put slightly less into savings in April than I do in March, why is that so bad? *I’m putting something into savings. *That’s awesome!

It is because I can feel, or at least feel the threat of, that negative second derivative. A negative second derivative—even for a function with a very high position or even one that’s increasing—means that the increase is slowing, or that the function is about to start tumbling down. A negative second derivative is, in fact, **a test for a function’s high point**. It has nowhere to go but down.

Here’s the other example, which matters much more than my financial stability. Things are getting worse for white Americans: there is apparently a finite amount of power to go around, and it is being redistributed (albeit at a glacial pace, unfortunately) to racial minorities. Despite still being on top, white America is falling. It could stop falling and *still *be in a better place than other racial groups in America, but that would not stop the fact that right now, white Americans feel a negative second derivative, and that can be scary. How we respond to the loss of our power is very important.

I think there are other examples that readers can find in their own lives, and it’s a useful idea to keep in mind: derivatives don’t say everything. Look around and see where you are, or so says your friend calculus.

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Now I have read the article. I’ll quote some of it here, which I’m sure is fair use on a two-post blog that has me about thirty-four dollars in the hole.

The

Was that quote too short? I’m sorry, that’s a consequence of the fact that **I just paid for a nine-paragraph article! **I may have the attention span of a millennial goldfish, but I make exceptions for the first time ever I bother paying my way past a paywall.

We’re not saying calculus shouldn’t be taught. — But the singular drive toward calculus in high school and college displaces other topics more important for today’s economy and society. Statistics, linear algebra and algorithmic thinking are not just useful for data scientists in Silicon Valley or researchers for the Human Genome Project. They are becoming vital to the way we think about manufacturing, finance, public health, politics and even journalism.

Linear algebra holds a privileged place in my heart, probably because it is the first time a student sees the “map that preserves an important quality” theme of mathematics. This is such an important feature of mathematical thinking that an adjunct instructor at a southern community college with a crappy blog might be willing to call it *the most *important feature, but he doesn’t want to get yelled at.

But why is it so important, Dr. Li and Professor Bishop (the former of whom runs a data science training institute)? I squirmed through a master’s degree without really learning statistics, and some attempts at learning topological data analysis (or even just regular data analysis) were thwarted by my 50-hour work week. It’s not obvious to me what relevance my appreciation for linear algebra as to do with understanding large data sets. (Otherwise I would be able to understand them.) (I realize I can and should educate myself, but I just read a persuasive article presuming to discuss the topic.)

My first point is this: unless you were already jonesing for a subscription to the *WSJ*, don’t pay for a nine-paragraph advertisement for learning data science.

I have a second point, and I think it’s much more interesting.

The *WSJ* article’s main point was that calculus was a critical body of knowledge in the space-and-missle-ridden 1900’s, but that in the information age it is less useful to students. This thesis supposes, as so many do but so few question, that the purpose of education is to get you a job.

Maybe it is. Education costs so much, whether in money, opportunity, or sanity, that it is not unreasonable for a student to expect a steady paycheck as a result of their hard work.

Maybe it isn’t. Maybe education, or at least the compulsory twelve years we all shuffle through in the United States, should be a public good designed to produce more well-rounded, more knowledgeable Americans.

In either case: who in God’s name needs to know how to evaluate an integral by trigonometric substitution? The answer is whoever tells WolframAlpha how to do it. Does the robot know how to do it now? Good, that first person can forget.

“But, idiot,” you say: “What if the computer forgets? What if all technology is destroyed in a freak EMP burst?” Right, and what if an airplane falls on me in front of this hookah bar? I should not have left my house today. We’d have bigger problems in such an unlikely scenario than losing a calculus trick, and if we still had the Pythagorean theorem, it would be fairly easily recovered anyway. (*Exercise: *Do you know how to explain trig substitution with just the Pythagorean theorem? It’s pretty neat.)

Look, I’m not saying that there’s not a class of people well-suited to preserving the ancient and mysterious art of crunching integrals and derivatives. I’m just saying that those people are not Every 18-Year-Old.

I believe calculus is incredible, and I believe as a subject it has something to offer *both *the utilitarian and hippy-dippy views of education I put forward at the beginning of this section. I do *not *believe—even if you are taking calculus to get an engineering job—that your life is improved by knowing the quotient rule.

Here are some interesting things that you could learn about in a calculus class. You could learn about how the limit was devised to answer a paradox posed by the Greek Zeno regarding motion, and how it can be applied to allow humans to use arbitrary small and large quantities. Without the limit, you’d try to invent a derivative that divides by zero and Newtonian physics wouldn’t exist. Instead, we take these arbitrary quantities that could as well be imaginary and apply them to solve real-world problems ranging from chemical decay to population growth to compound interest and *joke’s on you, dear reader,* those are the same thing anyway!

Calculus is a *stroke of human genius*, and we have relegated it to a required class in which the student comes away thinking that calculus is for slipping ladders and estimating how far away something is when you’re driving. (In the differential calculus course I TA’d as a graduate student, I asked my students for a freebie at the end to list some applications of calculus. One of my favorite students answered with the preceding. We are giving them the *wrong problems*.)

You can’t beat Something with Nothing, so goes the theorem of Republican game theory, and I am thinking of Something. This is where my article will sputter out and be ultimately unsatisfying, but at least I didn’t charge you $12.

Something here is a class, maybe called *General calculus *or *Calculus: history and application*. It consists of two pieces: whether these pieces are one, then a midterm, then the other or interwoven is currently unknown to me. The first act is the cultural and historical context in which calculus was birthed, all the times it was birthed: not just Newton and Leibniz, but Zeno and Weierstrass too. The second act is what calculus does for us now: *e.g. *optimization, arc length (required to be paired with a nod to Descartes’ claim we could never do it), and an *extremely healthy *understanding of differential equations. A student leaving this class would know that a differential equation is an equation whose unknown quantities are functions. They may even know how to solve a few. The A+ student would know how to grind one out by hand. The D student would know that chemical decay, population growth, and compound interest are all modeled by , and why.

My next move is to start poking around on Twitter to see if such a class exists, where, and how it’s doing. In the meantime, I want to ask a question: Why teach calculus? Because it is the cash cow on which all mathematics research rides around the pasture, waving its ten-gallon hat around and raving about the zeroes of something called the Riemann-Zeta function. Other than *that*, what’s the point? What do our students get out of it?

They will never die on a slipping ladder.

]]>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 . 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.

This post is for trigonometry students. At this point you know that an *angle *is a point on an origin-centered circle of radius 1, or the *unit circle*, and that cosine and sine are something that you do to those angles to get a number. You may also know that these numbers are the sides of the *right triangle* generated by the angle in question. *What*, exactly, cosine and sine describe is worth knowing, but I’d like to actually publish a complete post with a beginning and an end for once. Let’s get to the trick.

The *special angles *are points along the upper-right quadrant, or between and , that have “nice” values of cosine and sine. In other words, their values of cosine and sine may be represented in terms of fractions and square roots—things that are relatively easy to write down. These angles are

and .

The cosine and sine values of each special angle are listed in the table below:

This is, on its own, perfectly readable. As a trigonometry student, you know that cosine measures the “horizontal part” of angle: in other words, how much of the angle is moving left-and-right as opposed to up-and-down? You know that the angle is completely located on the horizontal axis, and so that it is a 100% horizontal angle. Therefore . Moving counter-clockwise, our angle is less horizontal and more vertical, so its cosine value shrinks. Drawing triangles and using the Pythagorean theorem told you that the next cosine values are and respectively. Finally, the angle is located on the vertical axis, so it is a 0% horizontal angle. That’s why . (Sine goes the same way, but backwards.)

That’s conceptual knowledge of cosine and sine, which I promised you wouldn’t need. Here’s the trick. Observe that and that . Now here’s how our table looks:

Now cosine proceeds by just subtracting one from the thing in the square root! Sine, conversely, adds one inside the square root with each special angle.

As you’ve learned, in the upper-left quadrant, cosine values become negative; in the bottom-left quadrant, both cosine and sine are negative; and in the bottom-right quadrant, sine is negative. With this, you know have a tool that produces the cosine and sine values for special angles as quickly as you can count. I’d go so far as to say you aren’t even memorizing anything!

I have some questions about the trick that may beget a future post (not to belabor the point, but I *really *need to get something written and out). Here’s one: What if we defined a curve such that the and coordinates of the curve are determined by which special angle you are on without the unit-circular condition ? What would the polar expression of said curve look like? (In other words, can we reverse-engineer the circle from this trick with the special angles? Can we do it with other curves?) Here’s another: can this trick be explained in such a way that ties counting up and down inside the square root with the conceptual truth behind cosine and sine? If we define special angles to be angles whose cosine and sine values have the form , could we modify that definition to yield different special angles?

Anyway, there’s the trick. Trigonometry is hard and memorization is a poor substitute for learning: at least now you’re thinking about cosine and sine ebbing and flowing with your position on the circle!

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