# Worksheet: Homology in Intro Lin Al

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?
• An excellent question. I don't teach at a high enough level to give you a teacher's perspective but perhaps I could offer you where my own difficulties have been with the topic as a student?
– Karl
Commented Aug 8, 2015 at 18:24
• That sounds great @Karl :) Commented Aug 8, 2015 at 18:25
• I was (am) uncomfortable with the vectors in the space. I find it challenging to understand what may be meant by a vector such as $5xa+xb+ab$ I could imagine a vector what could be traced around the triangle with your finger but that vector appears to warp to cover the same edge too many times. At this point I lose a sense of connection with the problem and become demotivated. Hope that makes sense?
– Karl
Commented Aug 8, 2015 at 18:35
• I find it difficult to understand your vector spaces without the picture of the triangle. Could you include it here? Are the points labelled $x,a,b$? If so, why the two different kinds of letters for the vertices? Commented Aug 8, 2015 at 19:31
• I don't know what homology is, so I am having trouble understanding what the title has to do with this exercise. Commented Aug 8, 2015 at 19:56

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

• Thank you for the story, and the sobering narrative. (TBH: I'm not entirely sure what your question was getting at, either: degree perhaps?) I didn't create this with any serious expectation that students would appreciate the homology— even after a touch of post-hw classtime devoted to it, which I think would help tremendously. More like, operating under the idea (delusion?) that the boring-ness of math class is really about routine, and using an unexpected, pictoral construction to combat that. Plus, it has the side benefit that the unusual construction actually comes from somewhere itself. Commented Aug 9, 2015 at 17:31
• The major takeaway from this should be that you won't know until you try it in the field. At the time I thought up the exercise, I was interested in testing students "moderately high-level" understanding of material they supposedly learned. The professor (who was considered an excellent educator and probably had more sense of the students capabilities than I in the course, a sophomore linear algebra course) had perhaps intended to teach me something about teaching. Regardless, I am grateful for the lesson. Gerhard "Teaching: Repeat Five Times, Individually" Paseman, 2015.08.09 Commented Aug 10, 2015 at 5:53

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."

• Thanks for the words of encouragement! Re 6: The other two examples are a square, and a square with the diagonal. Re 4: As much as I'd love to have tons of examples, the homework gets long very quickly. But a line segment is probably a necessity, even from purely LA perspective because it is rectangular "the other way". Maybe length-2 path, to avoid $n\times 1$ matrices. Re 1: This got brought up in the chat, and the more I think about it the more I like it, yeah. Commented Aug 9, 2015 at 1:13
• Re 6: As much as I want to move up a dimension, um, I don't know if it's reasonable to talk about quotient vector spaces. Though perhaps the "obviously artificial" nature of $\text{dim null } D_0 - \text{dim im } D_1$ would make it more obvious that this is a quantity they should be paying attention to, rather than coding "compute nullity" as "remember this one thing the prof mentioned in class" More seriously, the definition of $D_1$ is a lot less intuitive than $D_0$, and nontrivial $H_2$ would have to be done using 3D images which might get students stuck just in visualization. Commented Aug 9, 2015 at 1:21
• @EricStucky Ah, gotcha. I figured that this class would be at a level where quotient spaces would not be a problem. Makes sense not to move up in dimension in that case. Commented Aug 9, 2015 at 1:52
• Linear algebra doesn’t require the field of real numbers. Moreover, it doesn’t differ much across ground fields of characteristic 0. A matrix with integer components has the same rank and other properties over either ℚ, ℝ, ℂ, or whatever. The use of irrationals in examples would pointlessly distract students from the main idea. Commented Aug 14, 2015 at 17:19
• @IncnisMrsi I do not think it is such a big distraction. Homology over $\mathbb{Z}$ can have features like torsion which are not present over a field. Mainly, I just wanted to emphasize that if you are teaching a course about linear algebra over the field of real numbers (which is often what intro courses are), then your examples should contain some real scalars occasionally to emphasize that fact. Commented Aug 14, 2015 at 18:35

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 ;)

• Hey, sorry for the delayed response (life intervenes sometimes). Although it is a little more detail-oriented than I was asking for, I think that the other answerers have given me enough of a "green light" that this sort of advice is relevant ;) Thanks for your thoughts! Commented Aug 15, 2015 at 5:10