Why do we teach linear algebra in precalculus classes?

Question

When I took precalculus, we learned about polynomials and how to factor them, we learned about trigonometry and lots of great and useful identities there, and we learned about matrices. They didn't call it linear algebra, they just called it matrices. We learned how to multiply them, we learned a whole bunch of ridiculous complicated methods to calculate determinants. We learned to solve systems of linear equations using Cramer's Rule, which is the mathematical equivalent of scrubbing a floor with a toothbrush. Never once did we actually learn what matrices were, how they would be useful, or why we would want to know any of this. I found the whole exercise to be tedious and pointless and quickly braindumped all of the matrix stuff as soon as the class was over.

Now, 20 years, a degree in math, a degree in physics, a career as a software engineer and data scientist, using linear algebra in building neural networks and many other places later, I am teaching a precalculus class, and I still have no idea what the point of teaching about matrices in it is. Like maybe some basic matrix/vector multiplication would be useful, applying them to systems of linear equations, but why do we force our students to painfully calculate determinants by hand when literally no one does this and the only reason to study these algorithms is if you are implementing a computer program to calculate them.

Don't get me wrong, I am a huge proponent of learning fundamentals and practicing working through problems by hand before you give it to your calculator. It is important to understand the concepts behind what you are doing and how they applies to the world around you so you can use them to solve problems in applications and they aren't just pure abstractions. Except that doesn't really work for matrices because we never bother to explain what they are or how to apply them, even less so for determinants.

Linear algebra is a very useful field used quite a lot in physics, engineering, computer science, and any number of other areas. And it has nothing to do with calculus. When I took a linear algebra class, I had to relearn all of these fundamental operations anyway, except there weren't any lengthy exercises in calculating determinants by hand because no one ever actually does that outside of a precalculus class. I have students asking me why we are learning this and I honestly don't have a good answer to give them. Why is it there?

Answer 1

Vector algebra is a standard 3rd-semester calculus topic (e.g., see OpenStax Calculus 3, Ch. 2-3). This includes calculations of the dot product, cross product, and related values. Standard applications include calculating results of work, force, and business productions.

While this can be done purely algebraically (e.g., OpenStax does so), it's natural to express these operations in matrix notation, so calculus books may have some sections devoted to showing it in that form. Example on my bookshelf: this appears in Stein and Barcellos, Calculus and Analytic Geometry (5E, 1992), Sec. 12.5-12.7. Common exercises of the determinant and cross product include computing the areas of parallelograms, volumes of parallelipipeds, possibly crystallography, and torque. (In fact, calculation of parallelipiped volume from the end of the calculus sequence is one of the key half-dozen standard questions we use for assessment of our graduating math majors every year in my community college department.)

So presumably matrices appearing in a precalculus class are partly meant to set the stage for this use in calculus. Related, the precalculus course has covered solving linear equations (possibly by substitution and elimination), and it would be natural to show yet another option, the "next step", so to speak. And many of the students who take calculus will need linear algebra, so why not give them an introductory taste of all that content at the next level? (without calling the course something unwieldy like Preparation for Calculus, Linear Algebra, and Related Mid-Undergraduate Mathematical Topics, of course).

In addition, matrices also appear in the standard undergraduate discrete mathematics and data structures course texts as the basic way to represent general relations, graphs, and weighted graphs (and hence used in implementations for Euler circuits, Dijkstra's algorithm, Prim's and Kruskal's algorithms, etc.). Note this is a context in which matrices are not being used to represent linear transformations, so it's a case that highlights that's not a priori an essential part of the meaning of a matrix. In my own work, this is my own customary usage of matrices, for example.

So that's my understanding of the legacy of why those subjects have been included. If someone thinks that's poorly justified, or those topics have been squeezed out of those courses in recent years, then I could imagine coherent arguments for that.

Answer 2

The College Board made curriculum decisions for their new AP Precalculus course that align with sentiments you express. The course is divided into four units, where unit four is titled Functions Involving Parameters, Vectors, and Matrices. But unit 4 is not assessed on the exam. Here is an excerpt from their Course and Exam Description document:

Units 1, 2, and 3 topics comprise the content and conceptual understandings in which colleges and universities typically expect students to be proficient, in order to qualify for college credit and/or placement. Therefore, these topics are included on the AP Exam. Unit 4 consists of topics that teachers may include based on state or local requirements.

Answer 3

Because derivatives are linear operators.

Answer 4

Linear algebra is a very useful field used quite a lot in physics, engineering, computer science, and any number of other areas. And it has nothing to do with calculus.

It is incorrect to say linear algebra has nothing to do with calculus: linear algebra is all over the place in multivariable calculus. Maybe you meant to say that linear algebra has nothing to do with single-variable calculus.

I too saw some matrix algebra (in the $2 \times 2$ case) in my high school precalculus course. I never thought much about why anyone decided to put it there. Possibly it was included to expose students to some matrix work so the concept of a matrix would not be completely new if/when the students get to multivariable calculus. Or the person designing the curriculum knew that matrices are a mathematical tool that also comes up in later coursework not relying on calculus, e.g., some "math for business/economics majors" courses that I've seen have a unit on matrices: consider the Leontief input/output model.

You compare Cramer's rule to "scrubbing a floor with a toothbrush". For the purposes you saw of it, I understand that impression. But in higher mathematics (at the level only math majors would see), Cramer's rule is genuinely useful. It is one of the methods that implies algebraic numbers form a field, it shows ${\rm GL}_n(\mathbf R)$ is a topological group, and it shows ${\rm GL}_n(K)$ is an algebraic group for (algebraically closed) fields $K$.