College Algebra Exam Review 6

College Algebra Exam Review 6 - 16 1 ALGEBRAIC THEMES R and...

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Unformatted text preview: 16 1. ALGEBRAIC THEMES R, and T .x / D Ax C b is an invertible affine transformation of R3 , then T .v/ is a vertex of the convex set T .R/. 1.4.7. Let be symmetry of a convex subset of R3 . Show that maps vertices of R to vertices. (In particular, a symmetry of a rectangle or square maps vertices to vertices.) 1.4.8. Show that there are exactly six symmetries of an equilateral triangle. 1.5. Permutations Suppose that I put three identical objects in front of you on the table: This configuration has symmetry, regardless of the nature of the objects or their relative position, just because the objects are identical. If you glance away, I could switch the objects around, and when you look back you could not tell whether I had moved them. There is a remarkable insight here: Symmetry is not intrinsically a geometric concept. What are all the symmetries of the configuration of three objects? Any two objects can be switched while the third is left in place; there are three such symmetries. One object can be put in the place of a second, the second in the place of the third, and the third in the place of the first; There are two possibilities for such a rearrangement (corresponding to the two ways to traverse the vertices of a triangle). And there is the nonrearrangement, in that all the objects are left in place. So there are six symmetries in all. The symmetries of a configuration of identical objects are called permutations. What is the multiplication table for the set of six permutations of three objects? Before we can work this out, we have to devise some sort of bookkeeping system. Let’s number not the objects but the three positions they occupy. Then we can describe each symmetry by recording for each i , 1 Ä i Ä 3, the final position of the object that starts in position i . For example, the permutation that switches the objects in positions 1 and 3 and leaves the object in position 2 in place will be described by  à 123 : 321 ...
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