For my next installment on innovative teaching techniques, I’d like to dredge another demon that haunted my nights long ago—EIGENVECTORS! Normally, eigenvectors are introduced in the waning days of a first-year linear algebra course, when students’ minds are already saturated with new material and the stress of the later weeks of the semester, leading some to blindly memorize their theorems without really gaining a deep appreciation and understanding of them. Hence my recurring “eigenightmares” of sophomore year. One way around that pitfall is to introduce eigenvectors early on and then circle back to them throughout the course so students have a chance to use them again and again in a variety of contexts. This pedagogical strategy, known the spiral approach, can be applied to almost any skill or concept you wish to teach.
Let’s suppose your students have been multiplying matrices for a couple days and have just started to get the hang of it. Try presenting them with this matrix:
and this stick figure position on the coordinate plane as shown:
Remind students that an ordered pair, such as the location of the stick figure’s right foot, can be represented as a column vector. Have students individually or in a group multiply various key points on the stick figure by R to see what happens to the drawing.
Obviously the figure was stretched, and it should be obvious that the direction and magnitude of the stretch correspond to the diagonal entries in R. Point out as well that the angle of the figure’s left leg has changed from about 45o to about 30o, so in effect, those points have been rotated around the origin. Ask students to identify which points were not rotated at all, merely pulled outward from the origin.
Now repeat the process with this matrix:
The points on the x-axis will be stretched as before, but the figure’s left hand will leave the y-axis and move rightward as the entire figure appears to stretch diagonally and right rather than upward and right like before. Once again, ask students to identify which points experience no rotation. This time, it should be points on the x-axis and points on the line y=x (the figure’s crotch in this case). I like to compare matrix multiplication to stretching out the fabric of the x-y plane like a stretchy bed sheet. I imagine a pair of people standing on the left and right sides of bed sheet pulling in opposite directions, and another pair at oppose corners, pulling slightly less hard. As the bed sheet is stretched, the designs on the fabric get distorted in a particular way. Challenge students to identify in what two directions the x-y plane is being stretched and by what magnitude.
At the point, the students will have basically invented eigenvectors on their own—all you need to do is provide the name “eigen” and then let the students formulate a definition themselves. Here’s one possibility:
Def: An eigenvector is the direction in which a matrix stretches vectors through multiplication. The factor by which they are stretched in that direction is called an eigenvalue.
It’s fine for now if the definition is cast in geometric terms; it will give the students something concrete they can picture in their mind’s eye. Later in the course when you circle back to eigenvalues, students will have a clear foundation for what exactly they are and why they are important.
This might also be a great time to build on your students’ natural curiosity and encourage them to pose conjectures or ask questions. For instance, do all matrices have eigenvectors? How many? Does one of them always have to lie along the x- or y-axis? How can we find eigenvectors without having to make drawings? Have students jot down these questions on the inside cover of their notebooks or hang them up somewhere in the classroom where students can continue to ponder them as the semester advances.
At its most basic level, spiral teaching is about on circling back to the same concept in more depth once it’s had a few weeks or months to sink in. You introduce eigenvalues with a concrete example, give students some practice, and then return to them later on to develop theorems. But sparking students’ curiosity can make this technique even more successful. If, in the intervening weeks, students find themselves pondering these questions every time they open their notebooks or gaze absentmindedly at the wall above the chalkboard, they’ll remember the concepts better when you return to them. Some students may already have foreseen some the theorems you plan to introduce thanks to their conjectures, while others will be itching to know the answer to the questions.