# Why do we teach linear algebra in precalculus classes?

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?

• The precalculus connection is not clear to me. Are you questioning why these things are taught in precalculus? Would you be happy if they were taught elsewhere? Or are you questioning why they should be taught at all in any course? May 11, 2023 at 1:50
• Re: "we never bother to explain what they are" -- I'm curious what you think they "are", if something other than a rectangular array of values? May 11, 2023 at 5:46
• @DanielR.Collins linear transformations. 3Blue1Brown gives a far better explanation than I got in 4 years of college. youtu.be/fNk_zzaMoSs May 11, 2023 at 6:29
• @DanielR.Collins that's like saying numbers are just symbols on a chalkboard. Describing matrices as just rectangular arrays without any context for how they are used does a massive disservice to your students. May 11, 2023 at 16:22
• @JoelCroteau: "and any such small system is much more easily dealt with by Gaussian Elimination" Well, this is true for matrices whose entries are explicitly known (unless in the $2 \times 2$-case, where Cramer's rule is very easy in any case). However, when you have a matrix that depends on several parameters (as it occurs e.g. often in physics) Gaussian elimination tends to be very tedious and can lead to a messy case-by-case analysis: as one doesn't know which matrix entries that occur during the elimination are non-zero, one doesn't know in which steps one has to swap two rows. [...] May 12, 2023 at 8:44

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.

• Some previous familiarity with matrices and determinants is also helpful in introductory physics, electrical engineering, and mechanical engineering, courses. It's also one of the many latter-in-the-course topics that are often jettisoned due to time constraints (e.g. probability, math induction, rotation of coordinates and conics whose directrix or semi-major/minor axes or asymptotes are not parallel to the coordinate axes, partial fractions, etc.), but probably kept more often than the others because of Cramer's rule for solving $2 \times 2$ and $3 \times 3$ systems of linear equations. May 11, 2023 at 9:06
• Personally, I managed to graduate from college with a bachelor's of science, including a course in multivariable calculus, without anyone, at any point in my formal education, ever having communicated to me the interpretation of matrices as linear transformations of the plane. I had to learn it by watching a YouTube video. Presumably it was an instance of the you'll learn it next year problem, but it still amazes me that "how to calculate a determinant" was considered more important. May 11, 2023 at 19:36
• @Kevin: Matrices as linear transformations (actually, as representations of linear transformations, and not just of the plane) is a central topic in elementary linear algebra courses that physical science and some social science majors take (often in 2nd year; this being U.S. perspective) -- see "first level" in this MSE answer. Depending on the multivariable calculus course (here I mean the end of the elementary calculus sequence, and not advanced calculus), this aspect may be only be briefly mentioned or it may form a major theoretical arc. May 12, 2023 at 7:37
• Doesn’t linear algebra also directly intersect with solving differential equations? May 12, 2023 at 14:35
• "by the time the students get to differential equations, they are supposed to be proficient in a bunch of previous material, and they aren't." How to get past the "mystique" of Maths May 12, 2023 at 19:26

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.

Because derivatives are linear operators.

• +1 I think this is quite an important point (although the answer might be better - and better received - if you add a bit of elaboration). The claim that linear algebra "has nothing to do with calculus" in the question seems unwarranted since Jacobi matrices are important for calculus in several variables. May 12, 2023 at 8:42
• Also with the hate on Cramer's rule in the question, while I agree that it is not really a good practical tool for inverting matrices, there is a plethora of connections between subdeterminants and surface elements, the transformation behaviour of normals and so on. Maybe the question should rather be, why are those connections not better highlighted in this and in later classes?
– mlk
May 12, 2023 at 9:21

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 $$K$$.

• "higher mathematics at the level only math majors would see" is precisely the sort of thing we should not be forcing every high school student to learn. May 17, 2023 at 18:30
• @JoelCroteau I agree that those applications I mentioned are not something to be taught to high school students. I mentioned them only to point that Cramer’s rule is still relevant in higher math, rather than being something that has become totally obsolete for everyone. If Cramer’s rule were taught in HS, it should be to do something at that level. Many things in math can be thought about in a basic way and advanced way.
– KCd
May 17, 2023 at 22:46