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