Question

#1, 2, 3, 4

Problem 1 The linear transformation T : x + Cx for a vector x ERP is the composition of a rotation and a scaling if C is given as c=[. 0 0.5 -0.5 0 - [1] (1) Find the angle o of the rotation, where --<<, and the scale factor r. (2) If x without computing Cx, sketch x and the image of x under the transfor- mation T (rotation and scaling) in the RP plane. (3) You can consider the recurrence Xx+1 = Cxx, k = 0,1,2,..., as the repeated application of the transformation T. What is X2020 if Xo (4) What does the trajectory of the dynamical system Xs+1 = CXk, k = 0,1,2,.. look like, an attractor, a repeller, a saddle point or a spiral? [1]

Answer 1

c=ro To 0:5] -0.5 The rotation matrin fro the rotation by angle origin is given by A = Fosd - sind 7 to otherwise d about [Sind cosd Let o be the scaling factor. => 8A=c => [scose ->sind rcosp=0 and osince =-0.5 cost=0 C=0 0,5 1 - [ puise psol -0.5 S ise -> 8=4 because 322 2 osolen a> puise pro Sind must be Since roo and => 1 d=31 negative. We ɣsind =-0.5 2015 is rsin(311) -517.0.5 => -8-0.5 => 16 = 30 and 8 2005 Ž

3 : (2). 2 .X=(1,1) ->n - -2 k 1 3½ 2 -1/4 • (X=(1/) • AX=(1,-1) - 1 know (3) Wet that if A= 5 cost A = cose - Sind [sind cosa ] the A = [cashid) - Sin(no) costaty Sin(nd) (osnd) for all nen. >> cn = (a)= "A"=(€)" [cos(22cm) -Sin(son) Lsin(3n) cos(39) We have the recurrence Xxx1 = C C Xx ixo Xkti = CXk -) X- Xkti-Chalo 1% =[] - ск, ĈXari Exk2 (+1 Xn cnxo for all nen. cos( 31 2020 - Sin (31 2020 Sin (3r2020 cos (311 2020

:) Xpozo to J[1] =-=[ 22020 X9020 9200 [] X, - Akti all koŅ (4) 'kti 2k+1 li Since Akti is a rotation matrin for <i for all KEN, qk+ the dynamical system Ykt =CXk, is going inward. [:] and а spiral ic, lim XxH K-zoo