Pedagogical question about linear algebra

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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 $\mathbf Q$ is rationalizing a nonquadratic denominator (e.g., $1/(1 + \sqrt[3]{2} + 3\sqrt[3]{4})$ and one nice application of linear algebra over $\mathbf Z/2\mathbf Z$ 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".

Answer 6

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."

Answer 7

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.

Answer 8

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.

Answer 9

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.

Answer 10

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.

Answer 11

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.

Answer 12

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.

Answer 13

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.