![\"Screenshot](\"http:\/\/blogs.ams.org\/mathgradblog\/files\/2016\/04\/Screenshot-2016-04-06-04.39.48.png\")
<\/a><\/p>\nand this stick figure position on the coordinate plane as shown:<\/p>\n
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Remind students that an ordered pair, such as the location of the stick figure\u2019s right foot, can be represented as a column vector.\u00a0 Have students individually or in a group multiply various key points on the stick figure by R to see what happens to the drawing.<\/p>\n
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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.\u00a0 Point out as well that the angle of the figure\u2019s left leg has changed from about 45o<\/sup> to about 30o<\/sup>, so in effect, those points have been rotated around the origin.\u00a0 Ask students to identify which points were not rotated at all, merely pulled outward from the origin.<\/p>\n Now repeat the process with this matrix:<\/p>\n At the point, the students will have basically invented eigenvectors on their own\u2014all you need to do is provide the name \u201ceigen\u201d and then let the students formulate a definition themselves.\u00a0 Here\u2019s one possibility:<\/p>\n Def:<\/u> An eigenvector<\/strong> is the direction in which a matrix stretches vectors through multiplication.\u00a0 The factor by which they are stretched in that direction is called an eigenvalue.<\/strong><\/p>\n It\u2019s fine for now if the definition is cast in geometric terms; it will give the students something concrete they can picture in their mind\u2019s eye.\u00a0 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.<\/p>\n This might also be a great time to build on your students\u2019 natural curiosity and encourage them to pose conjectures or ask questions.\u00a0 For instance, do all matrices have eigenvectors? How many?\u00a0 Does one of them always have to lie along the x- or y-axis?\u00a0 How can we find eigenvectors without having to make drawings?\u00a0 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.<\/p>\n At its most basic level, spiral teaching is about on circling back to the same concept in more depth once it\u2019s had a few weeks or months to sink in.\u00a0 You introduce eigenvalues with a concrete example, give students some practice, and then return to them later on to develop theorems.\u00a0 But sparking students\u2019 curiosity can make this technique even more successful.\u00a0 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\u2019ll remember the concepts better when you return to them.\u00a0 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.<\/p>\nThe points on the x-axis will be stretched as before, but the figure\u2019s 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.\u00a0 Once again, ask students to identify which points experience no rotation.\u00a0 This time, it should be points on the x-axis and points on the line y=x (the figure\u2019s crotch in this case).\u00a0 I like to compare matrix multiplication to stretching out the fabric of the x-y plane like a stretchy bed sheet.\u00a0 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. \u00a0As the bed sheet is stretched, the designs on the fabric get distorted in a particular way.\u00a0 Challenge students to identify in what two directions the x-y plane is being stretched and by what magnitude.<\/p>\n