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College Algebra Exam Review 1

College Algebra Exam Review 1 - 1.4 SYMMETRIES AND MATRICES...

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1.4. SYMMETRIES AND MATRICES 11 Define r k D .r 1 / k for any positive integer k . Show that r k D r 3k D r m , where m is the unique element of f 0; 1; 2; 3 g such that m C k is divisible by 4. 1.3.3. Here is another way to list the symmetries of the square card that makes it easy to compute the products of symmetries quickly. (a) Verify that the four symmetries a; b; c; and d that exchange the top and bottom faces of the card are a; ra; r 2 a; and r 3 a , in some order. Which is which? Thus a complete list of the symmetries is f e; r; r 2 ; r 3 ; a; ra; r 2 a; r 3 a g : (b) Verify that ar D r 1 a D r 3 a: (c) Conclude that ar k D r k a for all integers k . (d) Show that these relations suffice to compute any product. 1.4. Symmetries and Matrices While looking at some examples, we have also been gradually refining our notion of a symmetry of a geometric figure. In fact, we are developing a mathematical model for a physical phenomenon — the symmetry of a physical object such as a ball or a brick or a card. So far, we have decided
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