Exercises51parenleftbigg14√2√3-14√2-14√2√3-14√214√2√3 +14√214√2√3-14√2parenrightbiggNote that it doesn’t matter about the order in this case.17. Find the matrix of the linear transformation which rotates every vector inR3counter clockwiseabout thezaxis when viewed from the positivezaxis through an angle of 30◦and then reflectsthrough thexyplane.a25xa45ya54z10001000-1cos(π6)-sin(π6)0sin(π6)cos(π6)0001=12√3-1201212√3000-118. Find the matrix for proju(v) whereu= (1,-2,3)T.Recall that proju(v) =(v,u)|u|2uand so the desired matrix hasithcolumn equal to proju(ei).Therefore, the matrix desired is1141-23-24-63-6919. Find the matrix for proju(v) whereu= (1,5,3)T.As in the above, the matrix is13515352515315920. Find the matrix for proju(v) whereu= (1,0,3)T.11010300030921. Show that the functionTudefined byTu(v)≡v-proju(v) is also a linear transformation.Tu(av+bw)=av+bw-(av+bw·u)|u|2u=av-a(v·u)|u|2u+bw-b(w·u)|u|2u=aTu(v) +bTu(w)22. Show that (v-proju(v),u) = 0 and conclude every vector inRncan be written as the sumof two vectors, one which is perpendicular and one which is parallel to the given vector.(v-proju(v),u) = (v,u)-parenleftBig(v·u)|u|2u,uparenrightBig= (v,u)-(v,u) = 0.Therefore,v=v-proju(v) + proju(v).The first is perpendicular touand the second is amultiple ofuso it is parallel tou.23. Here are some descriptions of functions mappingRntoRn.(a)Tmultiplies thejthcomponent ofxby a nonzero numberb.Saylor URL: The Saylor Foundation
52Exercises(b)Treplaces theithcomponent ofxwithbtimes thejthcomponent added to theithcomponent.(c)Tswitches two components.Show these functions are linear and describe their matrices.Each of these is an elementary matrix. The first is the elementary matrix which multiplies thejthdiagonal entry of the identity matrix byb. The second is the elementary matrix which takesbtimes thejthrow and adds to theithrow and the third is just the elementary matrix whichswitches theithand thejthrows where the two components are in theithandjthpositions.24. In Problem 23, sketch the effects of the linear transformations on the unit square inR2.Givea geometric description of an arbitrary invertible matrix in terms of products of matrices ofthese special matrices in Problem 23.This picture was done earlier. Now ifAis an arbitraryn×nmatrix, then a product of theseelementary matricesE1···Ephas the property thatE1···EpA=I.HenceAis the productof the inverse elementary matrices in the opposite order. Each of these is of the form in theabove problem.