Question

3. Let T : R2 + Rº be the rotation by 1/2 clockwise about the origin, and let S : R2 + R2 be the shear along the y-axis given by S(x,y) = (x,x+y). (You may assume that these are linear transformations.) (a) Write down, or compute, the standard matrix representations of T and S. (b) Use (a) to find the standard matrix representations of (i) SoT (T followed by S), and (ii) ToS (S followed by T). (c) Let C C Rº be the unit square whose vertices are (0,0), (1,0), (0, 1), and (1,1). Draw, on separate axes, the image of C under the transformations (i) SoT and (ii) T.S. clearly labelling each image. (Please draw large, clear diagrams.)

Answer 1

3 a) The standard matrix of rotation by o clocknuse i given by coso sino) 1-sino coso sn, Standard meltix of rotation toy clockwils about the ligin is given by A cost/ sint/ .ro 17 I sin 1/2 cos/ - o

Standard matrix of s. foot have shy (7) standard trasit B={[] [:} we have ]). [1] sf[i]): [i] standard matrix of s in alvenber 6) i) SoT (I followed by s) Istandard matrix of sol in P= A.B | PELO OBI i) Tos (s followed by I) 0 - BA 10- ſiolo17-01 Oli) sor{[%])= ][4]= [+] Then,

SD TOS TOS Jos Tos

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