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o] - (3 1. (Use format short, except when specified otherwise) Recall that a counter-clockwise rotation about the origin by angle defines a linear transformation from R2 to R2. As discussed in class, the standard matrix of this transformation is cos - sin 0] Ro= sin cos e (a) Enter A = Re for 0 = "/10 in MATLAB. (Note: you can type things like cos(pi/10) in MAT- LAB.) Then use it to rotate the vector v= counter-clockwise by an angle of T/10. (b) Let B = Re for 0 = 7/12. Use MATLAB to determine if AB = BA. (c) * What does the previous result say about how these two different rotation operations interact with each other? (d) The composition of two rotations is also a rotation. Let C = AB, and we shall determine the angle of rotation for the composition. We know this matrix has the form Re for some 0. One way to determine 0 is to extract the (1,1) entry and take its arccosine (this assumes 0 is between 0 and ). This can be accomplished in MATLAB by typing t = acos (C(1,1)). In this case, t is a rational multiple of . Determine t exactly by switching to format rat and computing t/pi. (e) Switch back to format short. The inverse of Re is R 9. Verify this in MATLAB for 0 = 7/10 by comparing inv(A) with R_R/10- (f) Reflection about a line through the origin in R2 is also a linear transformation. Let Lo denote the standard matrix for the reflection about the line through the origin that makes an angle of 0 with the positive 21-axis. For example, Lo is just reflection about the 21-axis, and is given by Lo -=[ :-) It can be shown that the matrix Lg is given by Lo = RLOR_. (To understand this, you must convince yourself that the triple composition on the right performs the desired reflection. Think about what that composition does to a vector in the direction of the line we are reflecting about, and also what it does to a vector perpendicular to this line.) Compute the matrix Le for 0= 7/10. (g) Determine if Lx/10L0 = LOLX/10- (h) It can be shown that the composition of two reflections is a rotation. Determine the angle of rotation of LX/10L0. (As before, present your answer as a rational multiple of 7.) [2 1 8] 2. (Use format rat) Let A = 6 4 3 2 5 5