question: How to do the bonus question? 2. Consider the linear operator T, on R2
ID: 3116322 • Letter: Q
Question
question: How to do the bonus question?
2. Consider the linear operator T, on R2 that is a reflection about the line y- x, and the linear operator T2 on R2 that is a counterclockwise rotation by 90° a) Calculate Th (e1) and T, (e2) and determine the standard matrix of T b) Determine the standard matrix of T2 by either calculating T2(e) and T2(e2) or by using the 1. standard matrix for a rotation and substituting the appropriate angle c) Determine the standard matrix of T3 obtained fromthe reflection) 7, followed by (the rotation) T2(careful with the order!). d) Find the image of u = (-5,2) under T3 (i.e., calculate T(i)) BONUS: exemplify this matrix operator T3 by adequately transforming a picture in PowerPoint or any other software of your choice.Explanation / Answer
2. The matrix representing the transformation for a reflection about the line y = mx is A =
(1-m2)/(1+m2)
2m/(1+m2)
2m/(1+m2)
(m2-1)/(1+m2)
When m = -1, the matrix representing the transformationT1 for a reflection about the line y = -x is A1 =
0
-1
-1
0
Also, the matrix representing the transformation for a counterclockwise rotation by an angle is B =
cos
-sin
sin
cos
When = 900, the matrix representing the transformationT2 for a counterclockwise rotation by an angle of 900 is A2 =
0
-1
1
0
(a).T1(e1) = A1e1 = (0,-1)T and T1(e2) = A1e2 = (-1,0)T. Also the standard matrix of T1 is A1=
0
-1
-1
0
(b).T2(e1)= A2e1= (0,1)T and T2(e2) = A2e2 = (-1,0)T. Also the standard matrix of T1 is A2=
0
-1
1
0
(c ). The standard matrix of T3 obtained from T1 followed by T2 is A3 = A2.A1=
1
0
0
-1
(d). The image of u under T3 is T3(u) = A3u = (-5,-2)T
(1-m2)/(1+m2)
2m/(1+m2)
2m/(1+m2)
(m2-1)/(1+m2)
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