Name: -sInstructions: No caulculators, phones, or other technology allowed. To receive full credit,you must ju.stify yotrr work. If I cannot read or understand your work, you witt not ea,rnmany points for that problern. If you need more space for a probl€ffi, use the back of thep&ge.1. (10 points) Let u- [1,-3,-3J, v: l-2,4,5], ffid'!rv: [o,t,a].(u) Find 3u - a(u * 2v).-= 3',-/L *' Lf u" e 8V: *w #. fi v = f-l , s,3] f-lt t\z)uoJ = [,5,*z\,-\il(b) Find the length of u * v.LA+v = t-l' 1,')ll ,n+-vll =(r) Find the value of a that makes the vectors v and w orthogonal.V 4 w r 4*7rn + Lf -t 5ar 3^" +YMath 0280 * Exam 1October 6, 2AL4Dr. Thomas EverestV'vrf ; O if &=(d) Find the angle between u a,rrd v.CcSS=.--f--Vlf ut ll llvllaI Soi,nLr ong Il-lS"ot,.' ,., '-'" ,46f*qhLeave your answer in the form cos 0 :. . .f-= il6NG'tLl= *- t' -1.?-F '=t--{rt J ts2.(10 points) Show that Ar [| |il is invertible, and use A-Lto solve the systemAx.: b, where x - F;] and to - | I] .L"'J L-lJRr+tR:R,'ii(z f t c D4[o r o[oo tx= A'tol R*-t(, f t s 4O I ..-> lD 3 -tlI Lo I Io'-l Ksrz?z f t S qil.*-4 I " r Icj L0 o r0 lf,:lj'[r s c)!l 4L; j ?f*t s/tLLo Iqlr? f :1l,AT q€t I| *l{)toooIoc)*tt,-.lo,'olrDID,j5 4 f rr I lo- j *4 l* rtotqFFr f t 5 q"*F--> lo r IL"o I7 € \*t It - I---Brl*3A-t =Ioo()e'* I t€FltdlIboAre Ll] , Lil , E] rinearry independenr or dependent?dependent, find a specific linear combination that produces the z,e;rs-D h^ve ^- n o nLn v ,'o{ 9o Lrt {(}Y1 73. (10 points)If they areDuab Ax[-tt3It tsLs z'iRsrR. f I[*o vector.'/E I*Yg {\*tszl*jooTu.-1y5Da-,"{ca=[,* o-,-h J, Y AA^t ,TrC-; 3V8 "fl,V= | ;fL4. (10 points) (u)Find ABC if A:A B C =r,'+LDIF fl ,B:;l :il,ffid e:[l B:l]37I-:l [l : :1Fr=l;f r oIz DL,*l*"LCtn 5 vK-(SolbJ )'B : f o .]Lo i ji^,1 Lt' t/BA :ilfoLI o * I? O FZ1o q(b) Give an example of two matrics, A and B, such that AB and BA both exist, butAB # BA,Ifto*:6"3A= rjAB = [:ilftB f BA5. (5 points) The vectors u : [|] and v : [jt] are non-parallel. Find an equation ofthe form ar * W + cz :0 that represents the span of u and v in IR3.IDoItDl,Lt;l7l,/gI,/Syls3aZt-FIE3zKs"To bu Lvt^sist e""t*' lv33ZgX''-F3v*7V*) +32(7l'{- {wlLr* [^r*** a-"zx +3t3L*O[l .'l {1,7B tt.l._---*-"**+Ir ]lo'+Lo IetricA+Ar(3 points (bomrs)) Shoriy that if A is a $quare matrix, then (A + A') is a syrnmmatrix.SqY*ff1*LrtL ,'f BT: B.J(n{ &r)t = Ar+ (Ar)t = Ar+A:( ArAt)t = N + Ar^^A & Ar ,s syw)n*trrL(4 points (bonus)) Show that for u and v in R',*.v: 1,'u * rrl l' - ltU -.rll''llou will receive no credit if you ttse specific vectors.ll rA+vl I '-- -L ll tA,-v lltq,\((**v)o(L^l-n) j=€ (^*v)h (HFv) )f wr,{ + zL.,v + n/u d "/^ + zr ev FL/ / /r-JLT L't e v j,t/lBONUS6.Bo"*7.-Lq=-L4Id.'€-btt.* -L€- ItIF* L4, G,