Math 417 homework 2 solutions(Given only for problems that are not straightforward computation; contactthe instructor if you still have questions about the others.)Section 2.1 problem 1A transformationT:R3→R3is linear only if it satisfiesT(0) = 0.Thetransformation is not linear. Forx1=x2=x3= 0 we gety2= 2, noty0= 0.However, the transformation can be writtenT(x) =020010020x1x2x3+020.Outside of algebra, adding a constant vector still counts as “linear”.Section 2.1 problem 1A linear transformationT:R3→R3must satisfyT(α·x) =α·xfor allα∈R.But here2T(101) = 2-111=-222which is not the same asT(2101) =T(202) =-242.Note: alternatively one could check that another property of linear transfor-mations,T(x+y) =T(x) +T(y),is not satisfied either.1
Section 2.1 problem 44The transformation is linear, with matrix0-a3a2a30-a1-a2a10.Section 2.2 problem 2Counterclockwise rotation around the origin by an angleαis multiplication bythe matrixcosα-sinαsinαcosαForα= 60◦we get12-√32√3212Section 2.2 problem 1043is a vector on the line. Its length is√42+ 32=√25 = 5, so aunitvectoron the line isu=u1u2=4/53/5.The projection matrix isu21u1u2u1u2u22=16/2512/2512/259/25=1251612129.The reflection matrix is2u21-12u1u22u1u22u22-1=7/2524/2524/25-7/25=12572424-7.