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:
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?