Alg1231W6T4 - Reflections

Sunday, 2 September 2018

6:56 PM

Machine generated alternative text: Figure 1: Reflection in the y-axis -4 given by O -2 4 Consider the transformation T : R which is reflection in the y-axis The map T can be represented by the matrix 1, -2 y 1 2 y -1 o Then: y iii) If T(v) 1 , then x 6 10 and T(w) 19 1 , then v -4 3 —9 O and y , then v) If we reflect the line y = 7m + 6 in the y-axis we get the line y 6 mm + b where O and b 6 1 Recall: the Maple notation for the vector is < 2

 

Machine generated alternative text: Figure: Reflection in the x-axis -2 which is reflection in the x-axis. Then Consider the transformation R : IR2 IR2 —4 —1 4 2 -2 -4 O x i) The map R can be represented by the matrix iii) If y iv) If we reflect the line y Recall: then x _4 O and y 9m + 3 in the x-axis we get the line y mat + b where and b 1 • the Maple notation for the matrix 3 1 • the Maple notation for the vector 2 2 is 4 1,2 112

 

Machine generated alternative text: Figure: Reflection in the origin -4 -2 2 O 2 x 4 -4 which is reflection in the origin [0, 0]. Consider the transformation S : IR2 IR Then and the matrix that represents S is In terms of the transformations T and R , which are reflections in the y-axis and x-axis respectively, we have [select all that apply] S(v) = S(v) S(v) — S(v) — S(v) Ifwe reflect the line y = 7x + 5 in the origin, then we get the line y ma: + b where

 

Machine generated alternative text: Figure: Reflection in the line y 6 -2 4 2 o -2 2 4 6 An important linear transformation in calculus is the reflection U in the line y = x, since this corresponds to taking the inverse of a function. For example and in general The matrix that represents U is This matrix has the property that .42 O fib, as do all 2 x 2 reflection matrices. Note: please use the Maple notation for a 2 x 2 when entering .42 Ifwe reflect the line y = 6x + 9 in the line y = x we get the line y = mm + b where and b -32 Recall: 1 • the Maple notation for the matrix 3 1 • the Maple notation for the vector 2 O 112 >,< 314 2 is 4 1,2

 

Machine generated alternative text: Consider the linear transformation Then and 1 Recall: the Maple notation for the vector is < 1,2 2 —3 5 4 5 4 5 3 5 1 By drawing a picture; and perhaps some experimentation, we can determine that V is a reflection about the line (Make sure to enter your answer as an equation in Maple syntax; for example If we reflect the line y = 8x + 7 using V, we get the line y ma: + b where 28/29 o and b -35/29

 

 

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