mid-sol - Math 303 Midterm Solutions Instructor: Matt DeVos...

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Unformatted text preview: Math 303 Midterm Solutions Instructor: Matt DeVos First Name (please print): Last Name (please print): SFU email ID: Signature: Problem Score Value 1 10 2 9 3 8 4 9 5 8 6 8 7 8 Total: 60 1 Problem 1. (10 points) Construct a Cayley table for the symmetry groups of the following objects. a) (5 points) There are just four symmetries: the identity , a rotation by which we denote r , reflec- tion about the horizontal axis, denoted h , and reflection through the vertical axis, denoted v . This gives the following Cayley table: r v h r v h r r h v v v h r h h v r This is the same symmetry group as the rectangle from HW3. 2 b) (5 points) The only symmetries of this object are the rotations about the center. If we let r denote rotation by 2 5 , then the symmetries are ,r,r 2 ,r 3 ,r 4 and the symmetry group is given by the following Cayley table: r r 2 r 3 r 4 r r 2 r 3 r 4 r r r 2 r 3 r 4 r 2 r 2 r 3 r 4 r r 3 r 3 r 4 r r 2 r 4 r 4 r r 2 r 3 3 Problem 2 (9 points) Which of the following are transformations of R 2 ? (Justify!) a) (3 points) f ( x,y ) = ( y,x ) This is a transformation since it has an inverse. Indeed, f- 1 = f since f f ( x,y ) = f ( f ( x,y )) = f ( y,x ) = ( x,y ) . b) (3 points) f ( x,y ) = ( x 2 , 2 y ) This is not a transformation since it is not injective, for instance, f (1 , 0) = (1 , 0) = f (- 1 , 0) . c) (3 points) f ( x,y ) = ( x + y,x- y ) The function f is the same as the linear function given by " x y # " 1 1 1- 1 #" x y # ....
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mid-sol - Math 303 Midterm Solutions Instructor: Matt DeVos...

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