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<Text-field style="Heading 1" layout="Heading 1">Outline</Text-field> 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bkZqcUZbb0Zdby1GZnE2OVEqcGxvdHRvb2xzRidGL0YyRjVGaXFGOkY8Rj5GQEZDRkZGSEZLRk1GT0ZRRlNGVUZYRlpGZm5GanFGW29GanQ= 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
<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field> This worksheet is intended for use in the beginning of an Abstract Algebra class to help students to visualize the results of rotations and reflections that return the transformed triangle to the same area occupied by the original triangle. 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
<Text-field style="Heading 1" layout="Heading 1">Rotations around the Origin</Text-field> 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
<Text-field style="Heading 2" layout="Heading 2"></Text-field> 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An important example discussed in the text is rotation around the origin by angle 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 radians. We see that this rotation takes [1, 0] to [cos(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), sin(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)] and takes [0, 1] to [-sin(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), cos(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)]. This allows us to produce a matrix that can be used for the rotation (these two images are placed into the matrix as its columns). Note alpha must be expressed in radians. 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 rotmat := alpha -> <<cos(alpha), sin(alpha)> | <-sin(alpha), cos(alpha)>>; 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 In an equilateral triangle there are three rotations that leave the triangle occupying the same area in the plane: 0 degrees, 120 degrees, and 240 degrees. Define these 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 Rot0 := rotmat(0); Rot120 := rotmat(2*Pi/3); Rot240 := rotmat(4*Pi/3); 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<Text-field style="Heading 1" layout="Heading 1">Reflections in a Line</Text-field> In an equilateral triangle there are three reflections that leave the triangle occupying the same area in the plane. They are the reflections in the lines containing an altitude of the equilateral triangle. We can position our triangle so that one of the altitudes lies along the y-axis. 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 A reflection in the y-axis takes [1, 0] to [-1,0] and takes [0, 1] to [0,1]. This allows us to produce a matrix that can be used for the reflection (these two images are placed into the matrix as its columns). VR := Matrix(<< -1 | 0 >, < 0 | 1 >>); 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 (The other two matrices have not been developed yet. It does not seem worth it when students can see the results thus far with matrices. It is much easier with plottools.) 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
<Text-field style="Heading 1" layout="Heading 1">Define and Display an Equilateral Triangle</Text-field>
<Text-field style="Heading 2" layout="Heading 2">1. Define each edge of the equilateral triangle. </Text-field> Matrix representation of each edge needed for matrix multiplication. ET1list := [<1,-sqrt(3)/3>,<0,2*sqrt(3)/3>]; ET2list := [<0,2*sqrt(3)/3>,<-1,-sqrt(3)/3>]; ET3list := [<-1,-sqrt(3)/3>,<1,-sqrt(3)/3>]; 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<Text-field style="Heading 2" layout="Heading 2">2. Display the triangle in initial position.</Text-field> 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 ET1 := pointplot(ET1list, view=[-2..2, -2..2],connect=true,color=cyan,thickness=5,axes=normal): ET2 := pointplot(ET2list, view=[-2..2, -2..2],connect=true,color=green,thickness=5,axes=normal): ET3 := pointplot(ET3list, view=[-2..2, -2..2],connect=true,color=red,thickness=5,axes=normal): display(ET1,ET2,ET3); NictSSdDVVJWRVNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JTckNyQkIiIiIiIhJCErJHAtTngmISM1NyQkRi5GLiQiK1IwcWE2ISIqLUknQ09MT1VSR0YlNiZJJFJHQkdGJUYzJCIqKysrKyIhIilGOy1JKlRISUNLTkVTU0dGJTYjIiImLUYkNiU3JEYyNyQkISIiRi5GLy1GODYmRjpGM0Y7RjNGPi1GJDYlNyRGRUYrLUY4NiZGOkY7RjNGM0Y+LUkqQVhFU1NUWUxFR0YlNiNJJ05PUk1BTEdGJS1JJVZJRVdHRiU2JDskISIjRi4kIiIjRi5GVg==
<Text-field style="Heading 1" layout="Heading 1">Linear Transformations in <Equation executable="false" style="Heading 1" input-equation="Typesetting:-mrow(Typesetting:-mi(""), Typesetting:-mrow(Typesetting:-mi(""), Typesetting:-msup(Typesetting:-mi("R"), Typesetting:-mn("2"), superscriptshift = "0"), Typesetting:-mi("")), Typesetting:-mi(""))">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</Equation></Text-field> Mappings of vector spaces that satisfy the linearity properties can be represented as multiplication by a matrix. We follow the text's convention, treating vectors as column vectors so that the matrix is always on the left and the vector is on the right. A := <<1,-2> | <0,1>>; b := <1,3>; "A b " =A.b; 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To see how multiplication by a matrix changes a figure, we apply the matrix to a list of vectors. multmatbylist :=(multmat, listofvecs)-> map((x,y)-> y.x,listofvecs,multmat):
<Text-field style="Heading 1" layout="Heading 1">Counterclockwise Rotations around the Origin</Text-field>
<Text-field style="Heading 2" layout="Heading 2">1. Rotation of 120 degrees</Text-field> Apply a rotation of 120 to each of the three edges of the triangle. R120_ET1_list := multmatbylist(Rot120,ET1list); R120_ET2_list := multmatbylist(Rot120,ET2list); R120_ET3_list := multmatbylist(Rot120,ET3list); 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Prepare plots and display triangle after 120 degree rotation. R120_ET1_edge := pointplot(R120_ET1_list, view=[-2..2, -2..2],connect=true,color=cyan,thickness=5,axes=normal): R120_ET2_edge := pointplot(R120_ET2_list, view=[-2..2, -2..2],connect=true,color=green,thickness=5,axes=normal): R120_ET3_edge := pointplot(R120_ET3_list, view=[-2..2, -2..2],connect=true,color=red,thickness=5,axes=normal): display(R120_ET1_edge,R120_ET2_edge,R120_ET3_edge); NictSSdDVVJWRVNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JTckNyQkIiIhRi0kIitSMHFhNiEiKjckJCEiIkYtJCErJHAtTngmISM1LUknQ09MT1VSR0YlNiZJJFJHQkdGJUYsJCIqKysrKyIhIilGOy1JKlRISUNLTkVTU0dGJTYjIiImLUYkNiU3JEYxNyQkIiIiRi1GNC1GODYmRjpGLEY7RixGPi1GJDYlNyRGRUYrLUY4NiZGOkY7RixGLEY+LUkqQVhFU1NUWUxFR0YlNiNJJ05PUk1BTEdGJS1JJVZJRVdHRiU2JDskISIjRi0kIiIjRi1GVg== Exercise: Copy, paste, and modify the above three sections of maple input to perform a rotation of 240 degrees and display the resulting figure. Apply a rotation of 240 to each of the three edges of the triangle. Prepare plots and display triangle after 240 degree rotation.
<Text-field style="Heading 2" layout="Heading 2">Reflections in the Lines containing theAltitudes </Text-field>
<Text-field style="Heading 3" layout="Heading 3">Reflection in the y-axis </Text-field> 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
Apply a reflection in the y-axis to each of the three edges of the triangle. VR_ET1_list := multmatbylist(VR,ET1list); VR_ET2_list := multmatbylist(VR,ET2list); VR_ET3_list := multmatbylist(VR,ET3list); 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Prepare plots and display triangle after reflection in y-axis. VR_ET1_edge := pointplot(VR_ET1_list, view=[-2..2, -2..2],connect=true,color=cyan,thickness=5,axes=normal): VR_ET2_edge := pointplot(VR_ET2_list, view=[-2..2, -2..2],connect=true,color=green,thickness=5,axes=normal): VR_ET3_edge := pointplot(VR_ET3_list, view=[-2..2, -2..2],connect=true,color=red,thickness=5,axes=normal): display(VR_ET1_edge,VR_ET2_edge,VR_ET3_edge); NictSSdDVVJWRVNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JTckNyQkISIiIiIhJCErJHAtTngmISM1NyQkRi5GLiQiK1IwcWE2ISIqLUknQ09MT1VSR0YlNiZJJFJHQkdGJUYzJCIqKysrKyIhIilGOy1JKlRISUNLTkVTU0dGJTYjIiImLUYkNiU3JEYyNyQkIiIiRi5GLy1GODYmRjpGM0Y7RjNGPi1GJDYlNyRGRUYrLUY4NiZGOkY7RjNGM0Y+LUkqQVhFU1NUWUxFR0YlNiNJJ05PUk1BTEdGJS1JJVZJRVdHRiU2JDskISIjRi4kIiIjRi5GVg== 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
<Text-field style="Heading 2" layout="Heading 2">Composition of Transformations </Text-field>
<Text-field style="Heading 3" layout="Heading 3">Example: Apply a rotation of 120 degrees followed a reflection in the y-axis. </Text-field> 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VR_R120_ET1_list := multmatbylist(VR,R120_ET1_list); VR_R120_ET2_list := multmatbylist(VR,R120_ET2_list); VR_R120_ET3_list := multmatbylist(VR,R120_ET3_list); 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Prepare plots and display triangle after rotation of 120 followed by reflection in y-axis. VR_R120_ET1_edge := pointplot(VR_R120_ET1_list, view=[-2..2, -2..2],connect=true,color=cyan,thickness=5,axes=normal): VR_R120_ET2_edge := pointplot(VR_R120_ET2_list, view=[-2..2, -2..2],connect=true,color=green,thickness=5,axes=normal): VR_R120_ET3_edge := pointplot(VR_R120_ET3_list, view=[-2..2, -2..2],connect=true,color=red,thickness=5,axes=normal): display(VR_R120_ET1_edge,VR_R120_ET2_edge,VR_R120_ET3_edge); NictSSdDVVJWRVNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JTckNyQkIiIhRi0kIitSMHFhNiEiKjckJCIiIkYtJCErJHAtTngmISM1LUknQ09MT1VSR0YlNiZJJFJHQkdGJUYsJCIqKysrKyIhIilGOy1JKlRISUNLTkVTU0dGJTYjIiImLUYkNiU3JEYxNyQkISIiRi1GNC1GODYmRjpGLEY7RixGPi1GJDYlNyRGRUYrLUY4NiZGOkY7RixGLEY+LUkqQVhFU1NUWUxFR0YlNiNJJ05PUk1BTEdGJS1JJVZJRVdHRiU2JDskISIjRi0kIiIjRi1GVg== 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
<Text-field style="Heading 2" layout="Heading 2">Comparison with results via plottools </Text-field> Done in a separate worksheet. 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