Worksheet: Homology in Intro Lin Al

Question

I am about to start grad school and I am trying to think seriously about teaching [you know, before I get swamped with my own coursework]. I wrote a hypothetical worksheet for an introductory linear algebra class which tries to do something a little more interesting than "Here is a random LT I made up, compute the nullspace". Here is an excerpt from the beginning:

Triangle (T). We will use the following graphic to create two vector spaces and a linear transformation between them:

$\qquad\qquad\qquad\qquad$

The vector space $C_0$ is given by formal linear combinations of the labelled points. In other words, $(x,a ,b )$ form a basis for $C_0$, so a typical vector in $C_0$ looks like

$$ 5x-3a -2b $$

Similarly, the vector space $C_1$ is given by formal linear combinations of the line segments. By analogy to the above construction, an explicit basis might be $(xa , ab, bx)$, so that a typical vector in $C_1$ looks like

$$ -6xa+2ab-8bx. $$

Finally, consider the linear transformation $D:C_1\to C_0 $ which takes an line segment $vw$ to $v-w$, if $v$ is the vertex the arrow is pointing toward, and $w$ is the vertex the arrow is pointing away from. For instance, $D(xa )=x-a$.

T1. Compute $D(-6xa+2bx-8ab)$.

T2. Write the matrix for $D$ using the two bases provided.

T3. By Gaussian elimination or otherwise, find all $\alpha\in C_1$ so that $D(\alpha)=5x-3a-2b$.

T4. Compute the nullspace of $D$ and its dimension (nullity).

Two more examples follow, all with "compute the nullity of $D$" ($4\times 4$ and $4\times 5$; a little large but not terrible) with the kicker at the end:

Further Thinking. Can you describe a relationship between the nullity of $D$ and the geometry of the graphic? Write a conjecture, using evidence (from your answers to the previous questions, or elsewhere) to justify it.

Ideally, this would be accompanied by a discussion of formal linear combinations in class, since the treatment above is clearly inadequate as a first introduction.

I came from an undergrad program with a high emphasis on STEM, so linear algebra education was unusual, as was the math motivation of the student body. I guess what I really want to know is, from people with actual experience in a classroom, would this fly? and if not, can it be salvaged?

But to make the question more amenable to the SE model:

• For what style/level of linear algebra course, if any, would this worksheet be appropriate?
• Are there obvious local improvements to what you see here? (i.e., could one of the questions be altered slightly to probe an additional concept)
• Can you point me toward education research that talks about the effectiveness of incorporating high-level ideas into lower-division courses?

Answer 1

I have a dim memory of a simple exercise I came up with as a TA. It was something like the vector space over the reals of polynomials in two variables, including x^2, y^2, and (x+y)^2. I then asked them to compute something like the dimension of the space S spanned by x^2 + y^2, (x+y)^2, and xy. I also asked them if xy was in S, and then if y was in S, and to explain why or why not.

The professor for the course thought it was a great problem, and that none of the students would understand it. He put it on a final exam with my name for attribution, and had me grade the result. He was right. A few students came close, but none were able to demonstrate full understanding of what was going on with the problem.

I think your exercise is good for an introductory graduate student or an ambitious undergraduate. Unless you have a top rate class though, the exercise will be under-appreciated. Or, it will take a long time to appreciate. This is not to discourage you so much as to alert you. If you turned this exercise into a lengthy blog post with some more detail worked out, you might reach a suitable audience.

Gerhard "Society Needs More Good Exercises" Paseman, 2015.08.08

Answer 2

Personally, I think this seems very appropriate. A few points:

1. You might reconsider using a two character name for a vector. $xa$ looks like a multiplication. Perhaps just label the vertices $V_1$, $V_2$, and $V_3$, and edges $E_1$, $E_2$ and $E_3$?

2. Since this is linear algebra, I assume you are using the reals as your field of coefficients. However, your example coefficients are all integers, which may give the wrong impression. You could throw in a $\sqrt{2}E_1$ somewhere.

3. Give some examples of vector operations in this space first, since it may not be completely obvious otherwise.

4. You might actually consider starting with the line segment, and seeing that its homology is trivial.

5. If your students have seen multivariable calculus already, it might actually be cool to do cohomology instead of homology, and connect this story to line integrals. I think Hatcher might actually motivate cohomology along these lines (the de Rham analogy)?

6. Obvious followup would be throwing in a couple more holes, or moving up a dimension. Also do not neglect "seeing" how anything that looks like a circle would have a one dimensional $1^{st} $ homology group. At least try a square as well.

7. I would include some kind of motivation, even if it is just "this is the very first step into a wide world of mathematics called Algebraic Topology. We aim to capture some qualitative features of shapes by computing algebraic quantities associated to those shapes."

Answer 3

I am a huge fan of using tools in a current math course to show something "flashy" or more advanced. It's especially nice if you can really ham it up and make people think they're doing something that no one else at their level is doing.

I was shocked when I learned that I knew enough to do this kind of stuff and was sad that I hadn't been shown it before! So I am impressed with your idea of going down this road.

A suggestion of usage. Obviously you would want to motivate this, perhaps along with dealing with formal linear combinations you might take Steven's idea 4 and do that yourself as a lead in for everyone to see. Definitely make sure that labeling is clear -- notation is the key to mathematical clarity!

Also, I would suggest taking the last 15-20 minutes of a class to have students begin working on this in pairs (or make it a paired homework). It is abstract enough to warrant team work, and sometimes requiring teamwork is useful. It would also be useful if you did it in class, because you could wander the room and field questions or listen in carefully to see where roadblocks are happening.

On an editing note, I would state the two vector spaces' and linear transformation's names in the first sentence. Repetition isn't bad.

(i.e.: We will use the following graphic to create two vector spaces, $C_0$ and $C_1$, and a linear transformation between them, D : $C_0 \to C_1$ )

All hypothetical, of course ;)