Why do some linear algebra courses focus on matrices rather than linear maps?

Question

I hope the essence of the question is clear from the title. There are obvious advantages to making the linear map the central notion of a linear algebra course:

• the notion can be illustrated with nice, intuitive geometric (or otherwise) examples, motivating the whole subject;
• the composition of maps is a very natural thing, from which the definition of multiplication of matrices flows smoothly;
• many results, such as the structure of a solution space to a linear system, of spectral theorem for self-adjoint operators, can be nicely visualized;
• many proofs (e.d., associativity/distributivity of the matrix product) become simple and conceptual, replacing unenlightening computations;
• the notion of determinant can be given its geometric meaning (or, if you dare, the true meaning as the action on top exterior degree), and the reason for the product rule becomes obvious;
• After all, that's just right, mathematically and physically: the primary object is the linear space and the linear map, the basis, and thus the matrix, is a matter of choice, secondary, technical devices.
• Many people actually complain that they "never get the linear algebra", that it's all abstract nonsense for them, so why not try and equip them with the way we, mathematicians, understand it?

Yet I have the impression that a typical undergrad linear algebra course, especially a "service" course, never goes this way. For example, Strang's textbook first introduces "the concept of linear map" on page 400, after most of the theory has been covered, almost as to make sure it's left off any actual course due to lack of time.

I can also see some reasons to actually prefer the matrix approach:

• the formal manipulation are easier to teach (and, especially, easier to grade);
• before, the students have been taught Math as a set of formal rules applied to numerical examples, so they are well prepared to be fed more of those and are unable to learn to reason with abstract notions such as "linear space" and "linear map".

However, neither of these reasons are satisfactory. We could say that if the students are unable to grasp these concepts, we would serve them better by training them in that, rather than teaching them the singular value decomposition without a proof. We could also say that if the conceptual understanding is so valuable for them, one should actually allocate some resources into that.

So, are there any actual reasons to prefer the matrix approach? Why is it so popular?

Edit: since many understood this post as advocating throwing matrices away altogether, let me tell that this is not the case. I admit that matrices are important. What I don't understand is the pedagogical value of, for example, exposing students to matrix multiplication before (or without ever) discussing compositions of linear maps.

Answer 1

Welcome Kostya! The mapping view is definitely important, but I don't think it's supreme. For me here's how I think about it. There are three ways to think about (basic) linear algebra: As a theory of matrices, linear maps and systems of linear equations. It is a fascinating result that when you formalize these three views the mathematical objects are isomorphic!

But each of the three views is useful for different kinds of applications, even very similar ones. If I'm trying to solve a system of linear equations, say to fix the constants for an IVP in a system of ODE's, I don't personally see how the linear transform view helps at all, whereas the systems of linear equations view is very useful. On the other hand, if I'm trying to construct a discrete dynamical system the mapping view is incredibly clear. If I put on my statisticians hat and try to fit

$\hat{y} = XW$

for $W$ by minimizing

$ RSS(\hat{y}) = (y - \hat{y})^T(y - \hat{y}) = \sum_{i=1}^N (y-\hat{y})^2 $

then it makes much more sense to think of $y$, $X$ and $W$ as matrices and vectors. Neither of the other views are immediately useful when defining matrix derivatives (unless we want to talk about differential forms).

Certain problems are easy to define under one view and not under others. I personally try to teach all three views so that students have a wider toolkit.

Now, sometimes with linear algebra there is a beautiful trick you can pull where you take a problem that is most clearly defined under one view and you switch to another view and extract some deep insight about its solution. Eigenvectors and linear maps are an excellent example of this. But it's the fluidity here that makes it useful, not the rigidity of fixing ourselves to a single view.

Answer 2

You might know (or not) enough computer science to know there are such things as functional programming languages. These are programming languages (the most popular are probably Scheme, ML, and Haskell) where loops and variables are avoided, computation is largely done by recursion, and, most importantly, functions are treated like data and can be passed as parameters to other functions. (For example, they have a library function, usually called something like 'map', which takes a unary function f and a list L as input, and returns a list whose contents are (in order) f(l) for each element l in L.)

This is an important and useful style of programming, but probably a majority of computer science departments spend less than 3 weeks (in their programming languages course) on it over an entire degree.

Why? They have accepted that most of their students will never be able to think of a function as an actual object. Their students will not be able to distinguish between a function $f$ and its abstract application $f(x)$, even after spending a few weeks working with this idea with a skilled instructor. (To be fair, our notation in calculus is terrible for this.)

You are way overestimating the cognitive abilities of an average person.

Answer 3

One reason is curriculum inertia—canonical math courses change slowly, perhaps best measured on a timescale of generations. For multiple centuries old material such as calculus and elementary algebra, the presentations have reached an equilibrium, but linear algebra is "new" enough that the education system is still figuring it out (the same thing can be said about discrete mathematics). Inertia is also how I would explain that synthetic geometry is still being taught to teachers, when all others have dropped it in favour of other material, but that's a digression.

@nate's point about three views of a common theory and fluidity of formalisation is definitely an important one in understanding the character of linear algebra—I for one always found least squares optimisation to be mysterious until I realised it was an orthogonal projection.

Some of the claims made in the question regarding the superiority of the map view should be nuanced:

• Matrix multiplication arises more naturally from matrix–vector multiplication than from composition of linear maps. One could of course argue that matrix–vector multiplication is applying a linear map, but that's an advanced view; quite often you organise a bunch of calculations as a matrix–vector product before you become aware that the vector can be thought of as living in a vector space. The take-home message for an educator is probably that a matrix should be thought of as something poised to act on a vector.

• Taking a geometric view of determinants doesn't require defining them primarily for linear maps, and in fact it can be an obstruction, even if $\det(AB)=\det(A)\det(B)$ follows immediately that way. The catch is that when starting from the linear map point of view you impose a distinction between rows and columns of a matrix—columns are geometric objects, rows are just arrays of numbers—whereas with determinants it is actually closer at hand to view the rows as geometric objects (at least if you wish to make a geometric interpretation of row operations, which you should: adding a multiple of a row to another row is just skewing the set whose volume the determinant measures, so of course the determinant remains unchanged). This is of course related to one downside of the linear map view you forgot to mention:

• The transpose is a whole lot more mysterious in the linear map view than in the array of numbers view. Abstractly the transpose can be understood as the contravariant functor mapping a vector space to its dual—if $A\colon V \longrightarrow W$ then $A^\mathrm{T}\colon W^* \longrightarrow V^*$ such that $A^\mathrm{T}(w^*)(v) = w^*\bigl( A(v) \bigr)$ for all $v \in V$ and $w^* \in W^*$—but that's really not the way one wants to describe it on a first linear algebra course.

Frankly, I find that the mainstream of linear algebra textbook authors are still struggling to free themselves of the historical bias that determinants are about solving linear equation systems.

Answer 4

As someone who is currently teaching an introductory linear algebra course and who likes matrices, let me tell you why.

First of all, here's why I like coordinates: if I want to tell you a specific position in the plane, the easiest way to do so is to tell you its coordinates. The idea that you can convert positions into tuples of numbers is extraordinarily powerful. It makes programming computers to do geometric things (ray tracing, flight simulation, anything) much easier. It's hard to imagine how modern algebraic geometry could have developed beyond lines and conics without this notion.

Now, why do I like matrices? Because they provide coordinates on the space of linear maps. Again, this is an extraordinarily powerful jump: just by writing down four numbers, I get a transformation of the plane. By writing down 25, I get a transformation of 5-dimensional space, which maybe I hadn't even imagined was a thing until now.

Without this device, even in 2D, I'd need to say something like "a shear by this amount with invariant direction pointing in the (17,63)-direction" or what have you. You may say that we never need to dirty our hands with explicit transformations like this, but in my research, when I think about almost toric Lagrangian fibrations and mutations of polygons, messy, explicit 2-by-2 matrices make my life an order of magnitude easier.

For students, this idea that you can encode a transformation with a finite list of numbers should be an empowering first step along the path to thinking of a transformation as a mathematical object in its own right. And the subsequent yoga of determinants and eigenthings give them the tools for manipulating these objects. You have to keep making the connection back to geometry so that they don't forget that what they're doing is more than symbol-crunching, but the symbol crunching helps them to internalise the idea that transformations are tangible things they can deal with.

If you start by telling them that a linear map is a function satisfying certain axioms, it will take them longer to figure out the domain in which they're working. Sure you can give them examples that are easy to explain (rotations, reflections, projections, whatnot) but the advantage of matrices is that you give them the potential to generate all examples for themselves and play with them.

composition of maps is a very natural thing, from which the definition of multiplication of matrices flows smoothly

I wholeheartedly agree, but I also think you don't need to introduce linear maps first to explain it this way. This is exactly how I define matrix multiplication in my course: once you've told them how a matrix defines a transformation, you can ask them to figure out the formula for the matrix defining the composition.

If you want to see the approach I take, my notes and videos are available here: http://jde27.uk/la

Answer 5

Context: I taught Linear Algebra as a junior level course for a mixed audience (math, engineers, science, education) who had already had a proof technique course and all three semesters of calculus including vectors and their analytic geometry. Of course, there were always a few math-education majors whose curriculum path forbade them to have the third semester of calculus (sad, these kids may well teach highschool math in the not too distant future, we really should require the math major for high school math instruction in the USA... I digress)

I try to teach with a balanced approach because I do see the merit in what you say. But, I also know that matrices are incredible useful and are truly interesting on their own. Here is the quick break-down of the course I taught a bunch of times:

• Week 1: what is the matrix ? Matrix addition, multiplication, matrix algebra, blocks and applications. I introduce all the basic component notation here and prove things like $(AB)^T=B^TA^T$. On a personal level, it gives me great joy for this reflects my personal interest in tensor calculation. Needless to say, students have mixed feelings about my exuberance for index notation. As the years went on, I find myself drifting more and more into column-based arguments. But, that is basically the take-away message notationally, to verify a matrix identity you can focus at the matrix, column or row, or component level. Each viewpoint has its merits.

• Week 2: Gaussian elimination and elementary matrices. I spend a day on row-reduction, a day on intepreting solution sets and a day on how row reductions can be implemented by left multiplication of an elementary matrix. I might slip an application in here somewhere. I do not attempt to "prove" the uniqueness, but I do emphasize the ideas of forward and backwards pass. If I had more time, I'd work LU-decomposition in here. In contrast to Week 1, almost everything here is a matrix or column level notation.

• Week 3: Inverse matrices, spanning, linear independence and the Column Correspondance Property (CCP). There is a good chunk of theoretical matrix algebra to cover here. I try to prove why left inverse implies right inverse in this context. The many equivalent characterizations of invertibility give a nice theorem to continue adding to as other ideas come in later. If I did Week 2 correctly, I've already showed them how to solve multiple systems with the same coefficient matrix so the usual magic trick for calculation the inverse is easily understood. Spanning and LI are new ideas, but the matrix calculations are the same we've been doing. Notice I focus attention here just on column matrix spanning and LI. The abstract version comes later.

• Week 4: Determinants motivated from volume. Laplace Expansion by minors and usual calculation tricks. Application to eigenvectors introduced (I have maybe one homework question on eigenvectors here just to prime the pump for later). Then, Week 4 ends with Quiz 1 and time for questions about the homework solutions which I provided.

• Week 5: Classroom interaction! (Test 1). Then we move on to abstract vector space definition and examples galore, subspace test and the theory of spanning and LI for abstract vector space. Many of my examples of abstract vector spaces are based on matrices. It's good they have lots of experience and we've already introduced the language to handle them efficiently $A = \sum_{i,j} A_{ij}E_{ij}$ etc. I introduce functions of vector spaces as an example of vector spaces. If you wish, I've introduced linear transformations here.

• Week 6: bases and coordinate maps, theory of dimension, linear transformations and their subspaces. I've tried various approaches over the years here. However, usually I use an argument which boils down to calculating the trace and using $tr(I_n)=n$ as well as $tr(CD)=tr(DC)$ to prove the number of elements in a basis is unique. There are many ways to get at this, and I always regret whatever I do since we could spend so much more time here to really get into all the methods. Also, by the end of the week I'm feeling guilty about all the properties of linear transformations I've forgotten to prove.

• Week 7: On restriction, extension and isomorphism. I try to impress on them how amazing it is to define a map on an infinity of points by its values on a handful of inputs. Linearity is hugely simplifying. The concept of defining a linear map by linear extension off a basis is introduced and used to formulate various isomorphisms. Then we return to linear transformations and introduce the concept of the matrix of a linear transformation. I typically spend a day showing how to calculate this in abstract case.

• Week 8: coordinate change for vectors and transformations. I draw pictures to derive the formulas then emphasize how these things simplify in special cases like column vectors or usage of the standard basis in $\mathbb{R}^n$. I have a few truly complicated examples in my notes which I'll project without actually working out. The point of the example is to share the motivation for the study: coordinate change allows us to find the most beautiful formulation of a given linear transformation. (incidentally, I think the under-emphasis of linear transformations in some curricula make coordinate change even harder to understand, even so, this is a difficult topic for most kids)

• Week 9: quotient vector space and the first isomorphism theorem, direct sum decompositions. (I try to prove things about cosets carefully and just sketch the idea of the direct sum and how invariant subspaces make the matrices nice). The depth of this week depends on the particular audience. I also try to take some time to contrast concept of null space for a matrix vs. kernel of a linear transformation. The coordinate maps are isomorphisms which transfer us between these various worlds.

Spring Break

• Week 10: Quiz 2 and Test 2, then the end of week we introduce Eigenvectors. Throughout the discussion of Eigenvectors I bounce back and forth between the e-vector of a matrix vs. e-vector of a linear transformation.

• Week 11: Eigenvectors continued, Jordan form. I don't prove everything here, although I do try to prove LI results about eigenvectors. I'll introduce notation for Jordan form and give examples, but I'm not going to show algorithmically how to find the basis nor prove its existence. Then I spend a day on complexification of a real vector space along with the concept of a complex eigenvector.

• Week 12: continuing from complexification we get the so-called Real Jordan Form which is what is actually needed to understand applications. Then, the rest of the week we dive into Inner Product Spaces and Euclidean Geometry. I try to spend a little time here talking about the various choices we have for norms and how the "circle" can be a square or diamond.

• Week 13: beauty of orthonormal bases, GS-algorithm, closest vector problem and orthogonal subspace theorems, application to least squares.

• Week 14: orthonormal diagonalization and the Spectral Theorem, sometimes I get into the proof of the Spectral Theorem, it depends how tired I am at this point.

• Week 15: application to real quadratic forms, application to calculus of many variables. Matrix exponential and solution to system of DEqns leveraging all we know about e-vectors, complex e-vectors and the real Jordan form.

• Week 16: multilinear algebra. Ok, I probably should cover the Singular Value Decomposition here, or the QR decomposition or something else. But, I should be allowed to have fun at least one week in a semester, right ?

In summary, I think your idea for teaching the course is fine, but you will still need to teach the matrix calculations somewhere because they come up in examples.

In any event, we should all teach from the heart. So, take your own advice before any of ours.

Comment following initial post on 5-18-2020 by Kostya:

...you start by introducing matrix multiplication, transposed matriced etc., by just imposing a set of formal rules upon students; then, presumably, drill the students on those rules... What is the pedagogical benefit of that? If you need to teach both matrix multiplication (which looks technical and poorly motivated to the beginner) and composition of linear maps (which is natural and very easy to motivate), why not first do the latter and then the former?

The pedagogical benefit of introducing notation is that it gives me a language which allows me to efficiently and smoothly communicate general examples. Matrix multiplication gives me a way to convert a system of linear equations into a single matrix equation. There is motivation from that alone for the matrix-column multiplication. hen, going beyond matrix-column vector products, $$ Ax_1=b_1, Ax_2=b_2 , \dots , Ax_s = b_s \Leftrightarrow A[x_1|x_2|\cdots|x_s]=[b_1|b_s|\cdots |b_s] $$ So, thinking about multiple systems of equations with the same coefficient matrix naturally leads to the concept of matrix multiplication.

To be honest, I don't motivate matrix multiplication when I define it. I just put it out there and start showing how it works. I take a more pragmatic approach, I do tell them that the initial definition is made so that matrix multiplication will fit with composition of linear maps. But, that is just a comment. I circle back to it later and show it explicitly once we later introduce linear maps. Then I circle back again later still and show it still makes sense with the extra baggage of coordinates ($T: V_{\beta} \rightarrow W_{\delta}$ and $S: W_{\delta} \rightarrow U_{\gamma}$ where $[T]_{\beta, \delta}$ and $[S]_{\delta, \gamma}$ gives $[S \circ T]_{\beta, \gamma} = [T]_{\beta, \delta}[S]_{\delta, \gamma}$)

So, yes, I do think students should be made aware that matrix multiplication can be defined by the necessity of adhering to the mechanics of function composition. But, on the other hand, I don't want to be talking about function composition while I'm focused on how to solve equations and interpret their solution sets.

Also, initially, I do want to share some enthusiasm for how we can use matrices to construct other objects. For example, typically I have them study the product of matrices of the form $\left[ \begin{array}{cc} a & -b \\ b & a \end{array} \right]$. Connecting that matrix with corresponding linear map would require much greater sophistication at this point in the discussion.

Answer 6

i like your approach but… it look on a way after whole journey—it’s probably quite difficult to imagine to you how you would approach the topic seeing it for a first time. You probably would have some nice geometric insight… for lines, planes, maybe for the space; this is somewhat enough to stop there (as e.g. cited strang generally stops at 2×2 matrices).

it’s beautiful to see all of those, whole picture; it is only a matter of where to start: i, my own, started with roughly systems of linear equations and jumped quickly to matrices, and rather stopped there trying to find what’s what geometrically later on.

this is the real question: where to start to provide most natural feeling, backstory, understanding, tools but also teach something more than mere knowledge—yet it be geometric or algebraic! in most courses there’re also other factors as time, courses attended so far, courses that depend on this knowledge (which part!?) etc. all this shall be taken into account when assessing ’which approach is best and why’—i’d really like to agree with you, i even sympathise with you but it’s rather obvious for me that you didn’t teach this topic from the very beginning any freshman! (don’t feel bad: me too! :D)

Answer 7

The text, "Linear Algebra Done Right" Springer(1997) by Sheldon Axler explicitly addressed the same problem. Matrices first appear in the last chapter (10).

Answer 8

Let me make the case that linear maps between vector spaces is the only way that linear algebra should be taught.

The rather old "definition" of a matrix is a set of symbols that "transform" according to a rule involving indices. But this "definition" is just the change of basis formula in a vector space for a linear map. The indices refer to a particular basis. Thus, the confusion - what "transforms" is the representation of a linear map with respect to the old and new basis.

Ditto for the old "definition" of a tensor.

The "definitions" are theorems - showing how the matrix representation of a linear map changes when the vector space basis is changed. The definition of a linear map is fundamental. The matrix representation is not.

Simmonds A Brief on Tensor Analysis has a side-by-side discussion of the coordinate-free representation (linear and multi-linear maps) and the older representation via indices. The coordinate-free discussion using vector spaces as the underlying algebraic structure is not only cleaner, but it is much easier to understand.