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#1, 2, 3, 4

Problem 1 The linear transformation T : x + Cx for a vector x € R2 is the composition of a rotation and a scaling if C is given as C-[ 0. 0 0.5 -0.5 0 [1] (1) Find the angle o of the rotation, where - <s, 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 R2 plane. (3) You can consider the recurrence X4+1 = Cxk, k0,1,2,..., as the repeated application of the transformation T. What is X2020 if Xo = (4) What does the trajectory of the dynamical system Xx+1 = CX, k = 0,1,2,..., look like, an attractor, a repeller, a saddle point or a spiral? [1]

Answer 1

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Cero 0.5 [si L-0.5 (1) The d about The rotation matrin for the rotation by aggle origin is given by A = Frost -sinf ] Sind cose Let or be the scaling factor. -> &a=c -> Focoseb - Yine 7 0,5 To 7 puise &cord -0.5 ÿ rcoso=0 and esind=-0.5 -> cose to beca ecause o to otherwise C=o. => 0 4 1 32 because o<6220 sind must be negative. Since roo and and rsindo => 1 0=31 3플 > -0.5 We rsind ==0.5 Es & sin(317) 80.5 ES -85-0.5 3.1.1 and r=0.5 = CamScanner

ya (2). 2 .X=(1,1) ->n -- / 32 2 •CX=(4,-) • Ax=(1,-1) Know (3) Wet that if A=cosch -Sind Isind cesto]the A9 = Ecotes Simenonj sin(nd) los mode) for all hEn. -) (n = A)" - "A"; M" A" = 4+ )" [(24) - Sin (30 -Sin(300) cos(397) We have the recurrence Xkti Xkti = (Xx = ²Xk- Exkl C XK x = [!] =ckt => Xx+1 = (kto Xn = ("xo -> =) 1202 X2620 = for all nen. (cos( 3n 2020) Sin ( 39 4020) cos(3n 2012 2 ber -Sin (3102020 2 CS Scanned with CamScanner

:-) Xpozo pravo to :] [:] ==[:] -) Xg020 ༢༠?o [] (4) Xkto a nem aka [:] Since A K+1 is and 2K+1 system Yxt =CXk a rotation matrin for all ken <i for all KEN the dynamical spiral going inward. [•] is a 1.C. lim XkH K-700 CS Scanned with CamScanner