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Last semester I taught a linear algebra class that is intended to introduce young students (at a sophmore-junior level) to "abstract mathematics". It seems that a major conceptual hurdle for many of the students is understanding the definition of a vector space. More specifically, a vector space is some set of things to which we can perform the operations of addition and scalar multiplication. Despite an enormous amount of effort on my part, many of the students insisted that it makes sense to do things like "take the real/imaginary part" of a vector or look at the components of a vector.

What strategies have you found useful for getting students to understand this type of definition?

I made this community wiki -- please edit it if the question seems badly phrased.

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Did you introduce the definition of a group or a field? –  Qiaochu Yuan Dec 20 '09 at 4:58
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Does MO have a policy yet on questions about teaching questions? We certainly will need one. PS this question isn't yet community wiki PPS sophomore-junior=2nd/3rd yr of university? 3rd year seems a little late to me to be learning what a vector space is. –  Peter McNamara Dec 20 '09 at 4:59
    
I've community wikified the question, since it looks like the OP wanted to but was having trouble with it. @Peter McNamara: What kind of policy did you have in mind? Why don't you start a new thread about it at tea.mathoverflow.net and post a link to it here. –  Anton Geraschenko Dec 20 '09 at 5:45
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Peter- Probably quite a few of the people taking this course are not math majors. I don't think there's a problem if economists wait for their 3rd year to learn about vector spaces. Also remember that American students are often not very focused in their first two years. Just taking the equivalent of 18.01,18.02, and 18.06 in the first two years would not be unusual, even for a math major (that was basically what I did). –  Ben Webster Jan 21 '10 at 0:17
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13 Answers 13

I think linear algebra is not a good topic to start with if you want to introduce students to abstract mathematics: because all n-dimensional vector spaces (over R, say) are isomorphic to R^n, it is not easy to say what has been gained by the abstraction. Of course, something definitely has been gained, but that something is hard to explain. With finite groups, by contrast, the role of abstraction is much more obvious: all you need to do is present two rather different groups (such as S_4 and the group of rotations of the cube) and show that they are isomorphic.

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IMO it's maybe not the best 1st course but it does make a good "redundancy" course for students to build-up confidence in their comprehension of abstract mathematics. –  Ryan Budney Dec 20 '09 at 8:17
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I agree. I think seeing what a group is, and what a homomorphism is before seeing what a vector space is, is a good way to go. I think that seeing group homomorphisms before working with vector spaces might help students understand linear maps, and especially help them to understand how to work with them without choosing a basis. –  Grétar Amazeen Dec 20 '09 at 15:26
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Gowers' remark is spot on. The gain from the abstraction - greater generality, no coordinates - becomes apparent when one studies, say, boundary value problems or the hairy ball theorem. I'd prefer to attempt a description of those things to beginning linear algebra students than take the disingenuous line of some textbooks, which claim that one gains from the abstract approach when all their proofs work equally well for subspaces of $\mathbb{R}^n$. –  Tim Perutz Dec 22 '09 at 12:44
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As I comment in my answer, I think that linear algebra is one of the topic where one has the biggest gain by abstraction. The very fact that for a given dimension you only have an example up to isomorphism means that with the minimun geometric effort (devising a proof for R^n) you get the maximum reward (everything you have done applies to polynomials, operators, matrices, solutions to linear differential equations, field extensions and so on). Many of these examples can be understood at the level of first year undergraduate. –  Andrea Ferretti May 27 '10 at 23:47
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I have addressed this issue at a slightly higher level, when teaching second or third quarter of abstract algebra for juniors and seniors. I had in mind a similar but not identical purpose, which was getting the students to truly understand the difference between a real and complex vector space. The best solution that I thought of was to teach something beyond that strictly requires an understanding of the difference.

Given a real vector space $V$, I define its complexification $V_\mathbb{C}$, and given a complex vector space $V$, I define its realification $V_\mathbb{R}$. Of course the conventional quick way to set up the former is with a tensor product, but without that scary idea, you can prosaically define $V_\mathbb{C}$ as $V \oplus iV$, two copies of $V$ with a defined multiplication by $i$. Meanwhile $V_\mathbb{R}$ is the same set as $V$, but with restricted scalars. These definitions can really get the students to think. They can consider that complexification inherits bases and does not change matrices, but does change the set of vectors. On the other hand, realification does not change the set of vectors, but doubles the dimension and requires extended bases. These issues are developed in very similar terms in Arnold, Ordinary differential equations, although in different notation. (I got the idea from Milnor and Stasheff.) Arnold has the very nice exercise of computing the realification of a complex matrix, and you can likewise ask what happens to the trace and the determinant.

Another pair of ideas that is helpful for overthrowing the idea of a set basis is, (a) the vector space of formal linear combinations of a set, and (b) a quotient of such a vector space by relations. A particular example is the color vector space and the reduced color vector space: $$\mathbb{R}[\{\text{red}, \text{green}, \text{blue}\}] \qquad \mathbb{R}[\{\text{red}, \text{green}, \text{blue}\}]/(\text{red}+\text{green}+\text{blue}).$$ The students cannot choose a basis of the second vector space without breaking a symmetry that they would like to keep. Expressing the same linear transformation of the reduced color vector space in different bases is enlightening.

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I would suggest the approach Tom Apostol takes in his linear algebra book. In chapter 1, after introducing abstract vector spaces, he goes on to Gram-Schmidt, and then immediately to best approximations. At the end of the first chapter, he solves questions like: "find the polynomial of degree three $p(x)$ which approximates $\sin(x)$ best over $[2,3]$ in the sense of minimizing the error $\int_2^3 (sin(x)-p(x))^2 dx$.

When I first read this, I was amazed. Prior to this, I only knew high school mathematics plus basic calculus - no abstract math at all. The problem of approximating one function by another seemed completely unsolvable given the mathematics I knew at the time. And yet here it had a simple solution.

Even more amazingly, the solution was right in front of me all along. If you had asked me how to approximate the vector $(1,2,3)$ by a vector whose last coordinate was $0$ - I would have immediately said $(1,2,0)$. I knew a little bit about geometry problems, and the problem of finding the closest point in a plane seemed "easy" and "natural" to me. And yet this this is all the solution of this problem required - all I needed was just to think about "vectors" or "points" a little more abstractly. I was completely sold on the benefits of the abstract approach.

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My understanding is that this is how calculators actually compute the sine, too. –  Qiaochu Yuan Jan 20 '10 at 21:54
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Qiaochu, I don't think that's actually true; that's just a lie we tell to calculus students to convince them that Taylor series are worth learning! Apparently the CORDIC algorithm (en.wikipedia.org/wiki/CORDIC) , which is based on calculating the sine or cosine by applying rotation matrices to the vector (1, 0), is what's actually used. –  Michael Lugo Jan 20 '10 at 22:16
    
Interesting. (Taylor series and L^2 approximations aren't the same, but I see your point.) –  Qiaochu Yuan Jan 20 '10 at 23:12
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I think the best way to appreciate abstraction is to actually see examples of vector spaces which are not $\mathbb{R}^n$ in any obvious way. For instance all polynomials of degree $k$ such that $p(0) = 0$ or all symmetric $3 \times 3$ matrices. For a more subtle example, subset of a finite set $X$ are a vector space over $\mathbb{Z}/(2)$ taking as $+$ the symmetric difference (until one realizes that this is just $(\mathbb{Z}/(2))^n$ in disguise). Students should be offered many exercises with these vector spaces, so that they become familiar.

When finite dimensionality is not necessary, one can make even better. For instance it is very nice to see the derivative as an example of a linear operator, and if one wants to have a finite dimensional example one can take the vector space of all solutions to some constant coefficients linear differential equation. Even a particular one, like $4f''' + 2f'' -f' -2 = 0$ will do. The derivative of a solution is a solution, hence we have a very natural linear endomorphism of this space. And by the way some linear algebra (for instance Jordan decomposition or even less) can be used to actually solve the equation.

Moreover I think that the fact that all vector spaces are isomorphic to $\mathbb{R}^n$ shows the power of abstraction at its best. Geometrically we only need to have the intuition for one very simple case; but then the proofs we give will apply to a plethora of other unexpected cases.

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I completely and utterly agress,Andrea.I personally think mathematicians use too many proofs and not enough examples in teaching-and if they don't give proofs,they turn the whole course into a Moore method problems course. I think a much more effective teaching philosophy is to give many,many detailed examples and state all but the most difficult proofs as either exercises or sketchy proofs. Theorums are generalizations of many examples and this method will emphasize this aspect of mathematics.It will also encourage them to build on this stock of examples,a very important skill for research. –  Andrew L May 28 '10 at 0:20
    
Yes, spending some time looking at the spaces of polynomials, matrices, and later their subspaces, describing bases, the dimension, doing Gram-Schmidt orthogonalization, etc really helps! I, too, like to do examples with symmetric/skew-symmetric matrices, as well as even/odd functions (or polynomials of bounded degree), and one can illustrate things like the kernel and image, and the rank-nullity theorem using the symmetrization map $A\mapsto (A+A^t)/2.$ –  Victor Protsak May 28 '10 at 5:20
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In addition to ODEs, which are fairly conventional (I even taught some courses with both LA and DE in the title), I'd like to advocate the use of $\textit{difference}$ equations: e.g. consider all sequences $(a_n), n\geq 0$ s.t. $a_{n+2}=a_{n+1}+a_n.$ They form a 2-dimensional vector space $A$ with shift operator $T: A\to A,\ (Ta)_n=a_{n+1}.$ Show how the Binet formula for Fibonacci sequence can be obtained "naturally" from the diagonalization of $T$. The book I used last fall (David Lay) spends some time on linear discrete dynamical systems, another good application of the same idea. –  Victor Protsak May 28 '10 at 5:29
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To understand a definition, show the students (a) lots of examples, (b) lots of non-examples and why they don't work, (c) misconceptions related to the definition (e.g., coordinates and real/imag. parts have no intrinsic meaning in an abstract vector space) and (d) applications which use the definition in a productive way.

In addition to showing a class how concepts they thought make sense in concrete settings (e.g., the first coordinate of a vector, or even that a vector has all positive coordinates) do not make sense in the abstract setting, show them that other things they have heard about do make sense abstractly, e.g., the determinant, characteristic polynomial, and eigenvalues look the same using two different bases, and those are the really important concepts. Otherwise they get the idea that all of their previous education in linear algebra doesn't work anymore.

You can't expect the students to catch on to the definition of an abstract vector space right away, but only over time, based on what you do with it. In the original question there was no indication of what was actually done with abstract vector spaces. A definition on its own will inspire few people, particularly a typical class of math majors with varying abilities. One nice application of linear algebra over Q is rationalizing a nonquadratic denominator (e.g., 1/(1 + 2^(1/3) + 3*4^(1/3)) and one nice application of linear algebra over Z/2 is the quadratic sieve factorization algorithm. These are not basis-free applications, but they serve to illustrate how the ideas of linear algebra can be used in settings that are not directly about "concrete vectors".

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This is something I could get behind, provided the coordinate free approach is stressed. (Not at the level of, say, Axler, who takes it way too far and somehow manages to lose the determinant entirely until the very end of the book.) –  Harry Gindi Jan 19 '10 at 18:04
    
I learned abstract vector spaces in a coordinate free fashion straight away in my first linear algebra course-but after we spent a month doing concrete calculations with matrices and thier determinants. I totally agree with KConrad in the importance of COUNTER-examples. They are not utilized enough in lower level courses and thier importance cannot be overestimated. –  Andrew L May 28 '10 at 1:14
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You could try giving the following example: the set of all positive real numbers, considered as a vector space over the field R, with vector addition given by multiplication and scalar multiplication given by taking exponents.

As a first step, you could verify that this satisfies a few of the vector-space axioms, and then let students check the rest of them (say, as homework). Then, you could ask questions like, "what is the dimension of this vector space?" or, "give an example of a (nontrivial) linear transformation from this space into R^3."

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I never saw the point of unnatural examples of vector spaces like this one, when in fact there are many natural ones: functions, polynomials, subspaces of other spaces, $\mathbb{C}$ as a real vector space, etc. This is just an example of "definition for definition's sake" mentioned elsewhere on this thread. –  Victor Protsak May 28 '10 at 4:52
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I disagree strongly with Victor. I think one of the problems I had when learning linear algebra was boredom: people claimed great generality of results but they were always applied to objects which always had extremely obvious representations as lists of numbers. Polynomials, especially if you're not going to multiply them together, are nothing but the list of their coefficients. Matrices are big lists of numbers. Everything that could "now" be treated just like R^n already looked exactly like R^n. SNORE. The multiplicative example is nice because it doesn't look "linear" at first glance. –  Pietro KC May 28 '10 at 7:49
    
Like Pietro says, by the time I could prove that all $n$-dimensional vector spaces "are" $\mathbb{R}^n$, this seemed apparent from the examples. To be honest, I was nevertheless impressed that our abstract definition led to such a concrete characterisation, and one matching the intuition, but this example would have seemed to give the theorem more content, and led me earlier to the interesting idea that $\mathbb{R}$ "is" $\mathbb{R}^+$. –  Max Nov 22 '10 at 22:31
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I'm not at all convinced that this is an unnatural example- is this example not where ln comes from? You can motivate exp and ln this way- Cartier does so in mat.univie.ac.at/~slc/wpapers/s44cartier1.pdf%20 (Thanks Tom Copeland!). You can argue, as Cartier does, that ln IS the vector space isomorphism from $\mathbb{R}^+$ to $\mathbb{R}$. This is the best mathematical motivation of ln that I know- "rate of growth" isn't mathematical, "integral of 1/x" is not compelling... but "converts * to + and exponentiation to scalar multiplication" is gold. –  Daniel Moskovich Apr 20 '12 at 5:46
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There is a substantial set of people who understand the notion of an interface in computer science, but don't understand abstraction in mathematics. For those people, it's worth pointing out that these are in fact the same thing (eg. if this is a linear algebra course for CS students).

Many programming languages (eg. the commonly taught Java) support the notion of an interface that is separate from an implementation. An example at wikipedia is that of a Predator interface shared by many different types of predator. CS students generally get the idea (or at least they should) that if they write for the Predator interface then their code can be reused with any predator, but that if they use implementation specific details of a particular predator then their code cannot be reused.

The situation is identical in mathematics. If you write mathematics for the "vector space interface" then your theorems can be reused for any vector space. But if you use specific knowledge of an underlying implementation (eg. about the specific set $\mathbb{R}^n$) then you lose the ability to reuse those theorems.

In fact, even if the class isn't being taught to CS students it's worth a brief mention as any large enough class of mathematics students is bound to contain a few who are computer savvy.

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Unfortunately, this still doesn't help too much if the student knows that every fin.-dim. vector space is isomorphic to some $\mathbb{R}^{n}$. To give a CS analogy, this is like writing a program that first determines what the specific implementation it is being applied to, and then uses implementation specific details to proceed. If it contains a comprehensive list of all possible implementations, then it can work quite well. Though it's a bit more subtle in the linear algebra case - we don't have access to the implementation, so we have to construct a virtual implementation isomorphic to it. –  darij grinberg Jan 23 '10 at 16:21
    
But still this doesn't explain to the student why it is that important to work at the interface level only. And I have no idea how to explain this, other than by using modules over rings (where not every module is free anymore, but lots of linear algebra don't hold either), by categories (the category of fin.-dim. vector spaces is not equivalent to that of the $\mathbb{R}^n$'s, or is it?), or by experience (superficial extra structure obfuscates stuff and makes thinking more difficult, and experience shows that vector space bases are often superficial, though a student will hardly believe it). –  darij grinberg Jan 23 '10 at 16:25
    
The category of f.d. vector spaces is in fact equivalent to the category whose objects are the non-negative integers and whose morphisms are matrices; the latter is the skeleton of the former. Categorically an important point to drive home is that in linear algebra there are related but different categories floating around, such as the category of f.d. vector spaces equipped with an inner product and so forth. (It is really really important to drive home that inner products constitute extra structure rather than a convenient calculation.) –  Qiaochu Yuan Jan 23 '10 at 17:20
    
Okay, how do you construct the equivalence between Vect^(fin) and the category of nonnegative integers and matrices? There is a canonical functor from the latter into the former, but how do you get the other direction? By a sort of axiom of choice for classes? –  darij grinberg Jan 28 '10 at 19:31
    
@darij: this is the well-known classification of equivalence of categories. an equivalence is just a fully-faithful essential-surjective functor. this uses axiom of choice for classes. –  Martin Brandenburg Jan 29 '10 at 13:01
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I can share what I did having a similar concern in mind, but it was for point-set topology, not linear algebra. I am not sure how much of this can be translated to linear algebra, since student's minds are already full of preconceptions about what a vector space, but not about what a topological space is.

After many years of tutoring point-set topology, I observed that students systematically thought of all topological spaces as $\mathbb{R}^n$, and that they always wanted to use balls, even if the topology was non-metrizable. Hence, when I got to teach my own point-set topology course, I tried something a bit radical: I did not talk about metric spaces at all until later in the course.

I started with motivation. On the second day, I defined the notions of topology, homeomorphism (but not continuous function), and convergence of a sequence. Then I did only small finite examples first. I gave the students the following exercise: 1) How many topologies can you define in {0,1,2}; 2) How many of them produce homeomorphic topological spaces?; and 3) In how many of them does the sequence $0,1,0,1,0,1, \ldots$ converge to $2$? Then I made sure to give students enough time (and guidance) to solve this exercise before moving to anything else.

I wanted to force the students to accept the abstract notion of topology and to not be scared by it (and to realize that everything we do in point-set topology is logical). Also, in this example, there is no way a student is going to attempt to use balls (particularly when I have not talked about balls). I think it worked well.

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To be honest, I would be somewhat confused by finite-set topologies - after all, they are not the raison-d-etre of topology, and they are not a particularly interesting object either (I mean, not even their number is known; to me this is evidence that they are not really in the spirit of mathematics). I would probably start with a parallel treatment of "real" topologies (the kind one has on $\mathbb{R}^n$ and on manifolds), of Zariski-style topologies (very important) and of $\mathfrak{m}$-adic topologies (such as on power series rings). –  darij grinberg Jan 23 '10 at 16:30
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Munkres has five main counterexamples in his topology course: R with the lower-limit topology, [0, 1] x [0, 1] in the dictionary order, the first uncountable ordinal, R^J in the product topology, R^J in the box topology. It's especially helpful to have examples of unfamiliar topologies defined on familiar spaces. –  Qiaochu Yuan Jan 23 '10 at 17:14
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Except that in this case, it leads us nowhere, or are there interesting nontrivial facts on finite-set topologies? Besides, the subtle but important difference between arbitrary (as in: arbitrary unions) and arbitrary finite (as in: arbitrary finite intersections) is lost in this finite example. Anyway, topology is not the subject by which people should be introduced to mathematics... –  darij grinberg Jan 28 '10 at 19:28
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@darij Every finite CW complex is weakly homotopy equivalent to a finite topological space. For example there is a space with 4 elements which has all the same homotopy groups as the circle. –  Steven Gubkin May 27 '10 at 14:48
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I think I'd rather risk that students use balls when inappropriate than having them puzzled at the meaning of this strange axioms of topology when they do not have even the most natural examples. There is time to grasp and appreciate the complexity. It sounds to me like introducing schemes without having seen quasiprojective varieties: maybe you will think twice before making a geometric argument which is not backed up by commutative algebra, but... well, unless you dump algebraic geometry altogether. –  Andrea Ferretti May 27 '10 at 23:39
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Every time that I've taught someone what a vector space is, I first spent some time on what a field is, including examples of finite fields. It's rather hard for someone to claim you can take the real and imaginary part of a vector if you say "Ahh, but the field I'm thinking of is $\mathbb{F}_2$, so what does that mean?" It's always seemed to help with what a vector space is to first see what a field is, and get some experience manipulating axioms that are more familiar.

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It might also be helpful to use counterexamples, e.g., $\mathbb{Z}$ is not a field under the usual operations...you don't have to say the words "ring", "module", etc., but I've personally always found that it helps to show where things can break. In mathematics you've always got the Scylla of showing that the objects of interest exist at all on the one hand, and the Charybdis of showing that said objects also aren't trivial or exhaustive. –  Steve Huntsman Dec 20 '09 at 5:10
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In Israel the two mandatory courses of first year undergrad math are real-analysis and abstract linear algebra (I think it's the same in Europe). You define fields before you define vector spaces, and you give as examples $\mathbb{F}_p, \mathbb{Q}, \mathbb{R}, \mathbb{C}$.

Once you teach what a linear transformation is, you have several examples involving $\mathbb{F}_2$ coming from computer science; e.g. Hamming code.

I'm not claiming that teaching first years abstract linear algebra is good (when I was an undergrad, half the students flunked first year math), just that if you do it you must have some non $\mathbb{R} / \mathbb{C}$ examples.

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That's the way it should be. –  Harry Gindi Dec 20 '09 at 17:26
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I'm not sure that's the way it should be (I'm not sure it shouldn't either). I taught linear algebra for science majors in Princeton; after one semester they've seen (with only partial proofs, and only over R/C) SVD, FFT and finite elements. Who knows more linear algebra: the student who knows what fields are and saw all the proof but not a single real life example, or the students who does not know what a field is, saw only few proofs, but knows important examples ? –  David Lehavi Dec 20 '09 at 17:58
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Real life examples are misleading. –  Harry Gindi Dec 22 '09 at 17:57
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@Harry: Bad real life examples are misleading. Good real life examples are e.g. the two body problem and pendulum (which started differential equations), heat conduction on a rod (which started Fourier transform), length of the lemniscate (which - via elliptic integrals - was the beginning of both algebraic geometry and a lot of complex analysis), three body problem (which started ergodic theory). They are anything but misleading. –  David Lehavi Dec 22 '09 at 20:38
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Time and again one meets people who have come up with the "proper way" to teach undergrads by introducing topoi before integers, and as many times one finds undergrads completely unresponsive to these methods. One suspects that, if such people got hold of a time machine, they would be utterly unable to teach their past selves anything. –  Pietro KC May 28 '10 at 8:08
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This is partly redundant with previous answers: one can present the students with a vector space that does not have a natural basis. What I would like to stress is that the simplest is probably the best, at least for a first example, and that the simplest is to take a vectorial plane in $\mathbb{R}^3$. What would be the two coordinates of a vector in $V=\{(x,y,z) | x+y+z=0 \}$?

I confess that students could be trying to think of these vectors in $\mathbb{R}^3$ rather than in $V$, so that this example maybe better to explain the need of a definition for a basis.

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This example actually makes a nice introduction to resolutions of vector spaces, since we get to keep the symmetry at little cost. –  John Wiltshire-Gordon Jul 2 '11 at 12:42
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Maybe I overlooked it, but I didn't see, in the previous answers, anything about a really geometric view of vectors. When introducing vector spaces, I like to use 2-dimensional vectors (arrows drawn on the blackboard, with the understanding that only length and direction matter, not the location on the board), with geometric definitions of addition and scalar multiplication. It is, of course, easy to explain that these geometric vectors are "really the same" as 2-component algebraic vectors (i.e., elements of $\mathbb R^2$), and also that the sameness depends on the choice of a coordinate system. This approach provides me with a lot of analogies for more complicated things that come up later in the course.

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The fixed ideas you describe probably originated from earlier calculus courses where students were exposed to "vectors" without any reference to vector spaces.

You could try some decontamination by first introducing groups, rings, fields and modules, before proceeding with vector spaces. Of course I do not suggest to turn the course into abstract algebra by going deep into group theory or ring theory; just giving a few definitions, plenty of examples and some immediate results, should be sufficient.

I see the following advantages:

  • the students meet something new right in the beginning, so they are less likely to fall into the "Oh, I already know this"-mode.

  • later definitions e.g. of a vector space can be build on previous ones and grouped into meaningful parts.

  • results like e.g. the homomorphism theorems can be given several times in slightly different situations.

The main problem with this approach is the danger of running out of time.

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