Having a glimpse of some of the possibilities for solutions of linear systems, we move to methods of finding these solutions. The basic idea we shall
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1 Hvn lps o so o t posslts or solutons o lnr systs, w ov to tos o nn ts solutons. T s w sll us s to try to sply t syst y lntn so o t vrls n so ts qutons. Tus, w rr to t to s lnton. T prry oprton nvolv s t ton o ultpl o on quton to notr. Expl Consr t syst 5x - y = 4 3x - 4y = -. Multplyn t rst quton y -3/5 n n tt to t son lnts t vrl x n t son n t syst os 5x - y = 4 -y = -6 Tn t son quton lrly rqurs tt y = 3 n, susttutn tt vlu or y nto t rst quton yls 5x - 6 = 4 so x = 6. Expl Nxt w xn t 3 y 3 syst low. x + 8y - z = 6 x + y + 7z = -3-4x - 33y + 5z = -3 T vrl x s lnt ro t son n tr qutons y n, rsptvly, - ½ n 4/ = ts quton on to t rsultn n t syst low. x + 8y - z = 6 y + z = -6 3y - 5z = - Fnlly, ultplyn quton two y - 3 n n to quton tr yls t ollown syst. x + 8y - z = 6 y + z = -6-5z = 7 Fro t tr quton t s lr tt z = -/3. Susttutn t vlu or z nto quton two n solvn w t y = -. Plun ts vlus or y n z nto t rst quton vs x /3 = 6, so x = 9 /3.
2 Dntons. T rst nonzro ntry n t row tt os t lnton s ll pvot.. T vlu o t ntry lnt v y t pvot s t ultplr. It s ultpl y t row w ontns t pvot w s tn sutrt ro t row ontnn t ntry lnt. 3. T or tt t syst s ru to s uppr trnulr. 4. T pross us to prou t soluton ro t uppr trnulr or s ll k susttuton. A syst otn ro notr y t ton o ultpl o on quton to notr s t s soluton st s t ornl syst. E soluton o t rst s lso soluton o t son n vs vrs. W sy ts systs r quvlnt. Intrnn qutons nturlly wll not n t soluton st o syst; so n quvlnt syst lso rsults ro ts prour. To v our ntrt ol o otnn n quvlnt uppr trnulr syst otn rqurs t us o quton ntrns s t xpls low sow. Prol Consr run t systs low to uppr trnulr or. Inty t nxt pvot n orrsponn ultplr(s). 8x - 4y = 3x + y = 3y + z = 6 5x + y - 5z = x - 6y + 9z = -8 7x + y - 4z = y + 8z = -3 y - 5z = 9 Expls Blow r uppr trnulr systs. Dsuss t or o t soluton o.
3 6x + 5y - 37z = 35 y + z = -4 6z = 5 3x - 78y + 34z = 6 3x - 78y + 34z = 6 3y + 8z = -49 3y + 8z = -49 = = 6 45x - 9y + 3z = 53z = = = - Exprssn t ont trx, A, o t nrl syst usn ts rows w wrot t syst wt ot prouts. W now tk ts oupl o stps urtr y rst lrn ts to t prout o trx n vtor,.. Ax = r x r x r x. It s portnt tt t ot prouts r n, tt t nur o oluns o A quls t nur o lnts (t nur o rows) o r r x. Sonly, w xtn ts to t prout o trs. Lt A = n B = (... p ) r wr t rows o A, t r k, n t oluns o B, t k, r vtors n R n. Tr trx
4 r r r r r r r r prout AB s n y AB = = (A A... A p ). An quvlnt r p p p wy o xprssn ts prout s y sttn AB = C = ( j ) wr C s t p trx wos,j t ntry j = j, t ot prout o t t row o A wt t j t olun o B. Expls ( 3) + 4( ) 4( 3) + 7( ) 6(5) + 4(8) = = 4(5) + 7(8) 6 ( ) + 43 ( ) 4 ( ) + 73 ( ) = Dntons. A squr trx M = ( j ) s onl trx j = j,.. ts ntrs tt r not on ts onl r zro.. T ntty trx s n n y n onl trx I n = ( j ) wt ons lon ts onl, = or ll. 3. A prutton trx P j s t ntty trx wt rows n j ntrn. Multplyn P j y trx A o t s sz to or t prout P j A ntrns t t n j t rows o A. 4. An lntry trx or lnton trx E j s trx w rs ro t ntty trx only n t,j t ntry. For ny trx A vn t s sz s E j, t prout E j A s t trx A xpt tt row j s ts row to t wr t,j t ntry o E j s.
5 Expls P = Lttn E = so tt t ultplr n t, poston s 3, w v 3 E = As w osrv rlr, t unt trx, A, o syst ontns ll o t norton n t syst. Also t rtl stps n solvn t syst nvolv only t two oprtons, row ton n ntrn, tt w n ow on A y ultpltons y E j n P j. W llustrt t prour on t 3 y 3 syst tt w solv rlr. Expl T unt trx s A =. For E = n E 3 = T prout E (E 3 A) = C s t unt trx o t son syst
6 3 w otn n run t syst. Tn lttn E 3 = U = E 3 C s t unt trx o t uppr trnulr syst. Sn t turns out s w sll s tt trx ultplton s ssotv, A(BC) = (AB)C wnvr ts prouts r n, w t LA = U wr L = E 3 E E 3. Cr os n to tkn sn trx ultplton s not outtv. W v not t prout o ts lnton trs L sn t s lowr trnulr n t prout o lowr trnulr trs.
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