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2 ivglentr zo ft Solving Linear Systems’ Aysb & ery > Ax=b nes ran ol mt Samescins X * equations Ler ¢ =rank (A) 4 ‘ auger (QB) SHE, reef ((AsB]) = CAF) Ye” mx(n+!) matin . consisterboage Yo inconsistent case ™ 4 0 o 0 leading 4 tm task column . Oo 4 i Tow. O14) Sunes “id ree yok 0 ‘ ' corresponds to equahon original system res S : i $0 4 Oxy + 1 O%H =A 0 o 4A +e or O=4 Or TS —> mr y Org _ treet zeo | 16 618 iN zero rows lead to +hvial anginal systern Ws equations 6 3 ¢ leading L's among an kam n columns ($0 ren) or r . Each of the first 1 columns corresponds to one of the variables x1,..,, Xn. Label the, leading entry column vanables as bound , the remaining n= variables as free, Fora given leading © 1, 19 agiven rove, the corresponding equation can be solved for the corresponding © ound variable (since it has a unit weffcient) th terms of the Free variables. Since each bound vanable only appears in one Spot in the reduced sysiem (zevols above and below Ths leading one coeffinent ) » Wedoes notenter into any other equation. Set the free variables equal 4 arbitrary constants ti ytaXee and bacesubsHtube these values tito the equations selved for the bound vanables, thus expressing all the varialles in terms of these “parameters, ‘er, “he solution is q paameinzed represerstation of the points the soluHon space of the system, terminolog y? different bocksuse different names” free variables = independent variables C freely specify > parameters) leading variables = bound variables = dependent variables (solve for ) also, changing the omer of the vanables changes their classifi cation EX. eqn for straight my-yab sos x-¥ek —~y { = bie + t1/m Ime, ya mxto or © vee ye A * su pes 2 differnt sdns yomna b (alrendy rlscod) nate y reef technique ? =bhamt depending on order & b y= b+ mtr eu) or Cys) where 62.5 bin + Un standard Form ort. = baht 
