Please solve parts e, f, and g. Thank you
(12 points - Matrix for reflection across a line in R2 ) In this problem, you will find the standard matrix for the linear transformation of reflection across a line in R2 that forms an angle of with the positive x-axis. Before doing this problem, go to Module 5 on linear transformations and watch Lecture 5 on Linear transformations in the plane. Pay careful attention to the last part, on projection onto a line that forms an angle of with the positive x-axis. You will do something very similar in this problem. (a) Write down the standard matrix for rotation radians clockwise. (b) Write down the standard matrix for reflection across the x-axis. (c) Write down the standard matrix for rotation radians counterclockwise. (d) Multiply the three standard matrices that you have written down above to get the standard matrix for reflection across the line that forms an angle of with the positive x-axis. Be careful of the order in which you multiply the matrices! (Hint: You can accomplish reflection across a line that forms an angle of with the positive x-axis in the following three steps. First, rotate radians clockwise so that the line coincides with the x-axis. Then reflect across the x-axis. Finally, rotate radians counterclockwise. ) (e) Check your answer by plugging in =0,=/4, and =/2 to verify that you get the standard matrices for reflection across the x-axis, the line y=x, and the y-axis, respectively. (f) Now suppose that the line has equation y=mx. Draw a picture to explain why makes an angle of =arctan(m) with the x-axis. Plug in =arctan(m) into your matrix from part (5d) and simplify to get the standard matrix for reflection across the line y=mx. Hint: draw a right triangle with legs of lengths 1,m, and 1+m2. The angle between the hypotenuse and the leg with length 1 measures radians. Use this picture to determine sin() and cos() in terms of m. (This is the same procedure as you use for trig substitution in calc 2.) (g) Using your answer to the previous part, find the standard matrix for reflection across the line y=2x. Using this standard matrix, find the coordinates for the reflection of the point (1,1) across the line y=2x. (12 points - Matrix for reflection across a line in R2 ) In this problem, you will find the standard matrix for the linear transformation of reflection across a line in R2 that forms an angle of with the positive x-axis. Before doing this problem, go to Module 5 on linear transformations and watch Lecture 5 on Linear transformations in the plane. Pay careful attention to the last part, on projection onto a line that forms an angle of with the positive x-axis. You will do something very similar in this problem. (a) Write down the standard matrix for rotation radians clockwise. (b) Write down the standard matrix for reflection across the x-axis. (c) Write down the standard matrix for rotation radians counterclockwise. (d) Multiply the three standard matrices that you have written down above to get the standard matrix for reflection across the line that forms an angle of with the positive x-axis. Be careful of the order in which you multiply the matrices! (Hint: You can accomplish reflection across a line that forms an angle of with the positive x-axis in the following three steps. First, rotate radians clockwise so that the line coincides with the x-axis. Then reflect across the x-axis. Finally, rotate radians counterclockwise. ) (e) Check your answer by plugging in =0,=/4, and =/2 to verify that you get the standard matrices for reflection across the x-axis, the line y=x, and the y-axis, respectively. (f) Now suppose that the line has equation y=mx. Draw a picture to explain why makes an angle of =arctan(m) with the x-axis. Plug in =arctan(m) into your matrix from part (5d) and simplify to get the standard matrix for reflection across the line y=mx. Hint: draw a right triangle with legs of lengths 1,m, and 1+m2. The angle between the hypotenuse and the leg with length 1 measures radians. Use this picture to determine sin() and cos() in terms of m. (This is the same procedure as you use for trig substitution in calc 2.) (g) Using your answer to the previous part, find the standard matrix for reflection across the line y=2x. Using this standard matrix, find the coordinates for the reflection of the point (1,1) across the line y=2x
