MAT 215 Final with Solution, Fall 2012

af ME 8 2) Let V be the vector space of all 2 x 2 matrices & h and let W be the subset of V Le d- consisting of all 2x2 matrices such that a~3h=0 and 2c+d =0. Determine whether W isa subs ce dy Pa 5 TY te A j : ‘ “ i! Pace of V or not. If Wis a subspace determine a basis and the dimension of W, otherwise (if it is not a subspace) skip this part. (10 points) 3) Solve the following system of equations by Gauss-Jordan elimination: xy +X = 2X, = 2304 le = BA 3x, +2x, —3x, —2x, Sat (13 points) te ieee =15 Soe ' = QF 4 49 8 ‘ i. SR Ra i: es Ae Ry x Ra—>R3 3 5 8 | 2 | ( co Sa: 4 ne Rs —> 83 ja rh ce oe 6 2 2. > URat Rr Rr eS — 4 2 ES —U~Rs4 Ry sR, oA aaa \ oe {a eel aa pany and he Gave Hee Free O oe ee 7 iA Variable s O ie aT ——aS™”-—S”—COrt—t 4) Let V = R2 with the operations: (@,,@2) + (by, bz) = (a1 + is at 22 a)» and Klay, a) = (ka, —k + 1, kag — 2k'+ 2). Show that V with these two operations salsly the first six axioms for a vector space. (10 points) we - @ Qet (04-7 RaNEY) V = (bey br) EN 7 pik) toe? dan! [O oN (or+by-') Qgtbr~2) EV. . eens Qi Vo (a4 b,- \y Q,4br-1 ); (ZaVaTa } >, =? — : = : 5 _ Cervair| ! br Part )= (Gari ieare pe (-Q\4 Ce . 8 —) “> "9 —_ Oat] Genet _ ed ca ee ¥v / ane wok =< (ajar) + ( : area aa e 6) Ke Ve — (kaye k rel kug-2k42) c V a let cr ey. coves to A3= \ (64) v7 = 5 “MG a e) Qe Ri Rr eels a Oi O.. 0 Slo | Wee ae) 6 % Ae! Se O => ‘ “ BaO As ae | Que b) Find a matrix p that diagonalizes, A, and determine the diagonal form of A, A” fi fOr any Positive ji, get — y rureey™ and evaluate 4” feo integer 7. (6 points) a + 22 — S ax 2k 2. = e) . on ) at Ry < R37 S i i ( a” Yo © * 0) ; ik ) oO Wer 'y,%— - \> ' o\ ha. e327 =e | Lega Vig re) Y, oe —_ - scoape . pay \ 7 \P = \ei ¥ (22 ~ 5 Y O =? \ > A \ 2 > rae | 0 Va A \ \ = = —f. 43 ee \ = Gr = 4 Ce > G e LEC O A’ - ant C Pos ae Pe 1.4 3 © ae “S rt Y De cas pee ; 2> ¥; form a basis for R? (8 Points) _ iy) Fe 2, ve. eo kG dtl — 1 o — eee Pal ee. ey y We } i x a ene set nem bh a of Rectovs = |S = Ss (R>) > and write v, as a linear combination of f 6) (Bic (oes. 1 / vf . » 0), v, = (—1, 0 1): y ah 2 oO), r : Show that v. ; » 1, 2),'v, =(—3, 2,1) be 4 vectors in R? . a Nas re) n a) Show that if a matrix A is row e elementary TOW operations), the quivalent to a matrix B (i.e. one can obtain B from A by n there exists an invertible matrix Q such that QA = B mentary matrices) 4 A wy YOu equivaden | So B Ren, ie ve are (5 points) chemen oo matrices ee ase ee puch yea RQ = EK Es ‘ns ec, = A q ee ee E., Ee c-- 5 Eve are clementory - a . ee cet : =? Ex. Ei < =e E2. eA VN TrAve kh We oe ts WVNECd ble bake . Qe Ex Gees Be Es. b) Show that if 4 can be written as the product of elementary matrices, then the linear system Ax =0 has only the trivial solution. (5 points) TEA canbe wetter as product Al shemen tory makrces Rea S J aderkibley => APP has o wh we goluka MONK We Laval colic <2 3 . ! oa ras eh 6. bes — — o t (5 points) al 5 a ~ = : aren : en —- ae (a Se »S) saben Oo a = (Be deb ((n. tJ") ee a ye (A- \t) Dy is vo (Catyt= Tt <> (0 \* Aa estat =e —> cA“) A ie aad eal e> (At). OF > fo Ww symmetric. C) The product of two invertible matrices (of same size) is invertible, (5 points) SG ae ee p va (ag) (@'A)= ATA =T (Q" Ar‘) (AR\= Bak. 7 “ar errs + SES rule to Solve the } ineae es Near system, (10 points) tt ZY 4g ee oy ee aa 37 — 9 |: aN LS a 4 Biche r\ a Vacs: | ats ..) MD gate tS 3 <3 det (A) ; — Jee Fe dek ite) de t(A) deta) | Vet(A) a deE(A— WC 6-2) -2 (6) 404-2) = Woop a= de (A) = 5 (6-2) -2 (18-9)41 (12-18) = o-\8—6 2-4 Jek(Aj= t¢1e—-9) —5(6-\) 4.1 -c)- 9-29 ANE de t{A,\= (19 APR Os (4) 5 6 GA 3 7 I (y= '| [ze2\ — cls