1 24120201 t 5 solutionsets ofl.me ms

AI I homogeneoussystem 5 8 EIR always a solutionT trivial solutionmxn matrix

Q are there non trivial solutions

AnswerHomogeneousequationAT has a non trivialsolution iff the eg hasatleastonefreevariable

EII 2x Xr 13 3 0Q is there a nontrivial solution2x 12 2 3 3 0

describetheset setGX 9 3 0

Solution

I o Is E I REFinelfmat X Xz XzT

freevariable

If I I I therearenontriusolutions

o O J X freeRREF solvefor

basicvars

I 1 g x x I solutionset is thelineSol invectorform

Note anontriu solution a a

4 13 R

2 SX Xz 6 3 0 Q describethesolutionsetsystem ofLeg on 3 vars

ImplicitdeSI X basic Xz Xs free Solve for X i X 5 2 2 3 sorptionXzfree oftheplaneXs free

Explicit parametricvectorGeneralsolution I 1 3 xz Its xs q term oftheplaneF f I Stettin St EIR

parameters

Thus Solset is Span ii I a plane in IRS

Solset to anyhomeyeg A I has theform Spanfit Ivp forsuitable vectors Th Jp C IRSpecial case setset Span I I case whenthere are no non tin solutions

Solutionsofnon homogeneoussystems

Exi Describe all solutionsof AI D with

f EmatrixofcoefficientsfromEx 1

east Hi 1 I x.IE itxso

F TSol set I p t j t c IR Recall solutionset of A I 5

in parametric TD was Iform ExsaddBto agenerics1 Same

p'ttv A 5 of AE Eaddition translationt Solution set of homeg eg AI J a linethroughAxes

y theoriginnonhomogeg AI 5 aparallel line

through Fex

theorem Suppose the eq A Esb is consistent for somegiven5and let p be a solution Then the set set of A I 5 is the set of vectors

of theform I p Eh where Ih is anyset ofthehemog eg AI

Note If is inconsistent the setset is empty Thmappliesonlyto consistentegs

writing a set set in parametricform thealgorithm

Aug Mat RR EFsolve for basic vars in terms offree vars ifanywrite a general Sol I as avector ul entriesdependingon free varsdecompose I into a Pin comb of vectors withnumeric entrees withfreewars

as parameters