Problem 7. (2+2+2+2-8 points) (a) Let S : R2 R2 be the linear transformation tha
ID: 3117448 • Letter: P
Question
Problem 7. (2+2+2+2-8 points) (a) Let S : R2 R2 be the linear transformation that reflects points through the (b) Let R : R2 R2 be the linear transformation which has eigenvalues 1 and , respectively. Find the line y r. Find the standard matrix of S and -1, with corresponding eigenvectors standard matrix of R (c) Let T : R2 R2 be the linear transformation T-SOR, that is, T(x) S(R(x)). Find the standard matrix of T (d) Explain why T is a rotation, and find tan , where is the (counterclockwise) rotation angle. Problem 8. (2+2+2-6 points)Explanation / Answer
7. (a).The matrix representing the linear transformation that reflects points through the line y = mx is A =
(1-m2)/(1+m2)
2m/(1+m2)
2m/(1+m2)
-(1-m2)/ (1+m2)
Here, m = 1, so that the required matrix is A=
0
1
1
0
(b). Let the standard matrix of R be M =
a
b
c
d
Since the eigenvector of M associated wth its eigenvalue 1 is (2,1)T,hence M(2,1)T = (2,1)Tor,2a+b= 2 …(1), and 2c+d = 1…(2). Similarly, since the eigenvector of M associated wth its eigenvalue -1 is (-1,2)T, hence M(-1,2)T = -1(-1,2)T =(1,-2)T or, a-2b= 1…(3) and c-2d = -2…(4). On solving these equations, we get a=1,b =0, c = 0 and d = 1. Hence, the standard matrix of R is M =
1
0
0
1
(c ). If T = SoR, then the standard matrix of T is AM =
0
1
1
0
(d).The matrix representing a counter clockwise rotation in R2, by an angle is
cos
-sin
sin
cos
so that T represent a rotation. Also, cos = 0 and sin = +1 or -1 Also, tan = sin/cos is indeterminate.
(1-m2)/(1+m2)
2m/(1+m2)
2m/(1+m2)
-(1-m2)/ (1+m2)