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Re: Strange empty set of solutions
*To*: mathgroup at smc.vnet.net
*Subject*: [mg71813] Re: Strange empty set of solutions
*From*: José Carlos Santos <jcsantos at fc.up.pt>
*Date*: Thu, 30 Nov 2006 06:05:03 -0500 (EST)
*References*: <ekh7pg$sgs$1@smc.vnet.net> <ekjfnl$d13$1@smc.vnet.net>
On 29-11-2006 8:17, Jean-Marc Gulliet wrote:
>> I have a certain 3 x 3 numerical matrix M. If I type
>>
>> Solve[M.{a,b,c}=={0,0,0},{a,b,c}]
>>
>> I get
>>
>> {{a -> 0. + 0.0410026 c, b -> 0. + 1.35294 c}}
>>
>> However, if I type
>>
>> Solve[{M.{a,b,c}=={0,0,0},a^2+b^2+c^2==1},{a,b,c}]
>>
>> I get the empty set. Why is that?
>
> According to the online help, "Solve gives {} if there are no possible
> solutions to the equations. "
>
>> I should get two solutions!
>
> Not knowing your matrix M, I cannot tell whether your expectation are
> legitimate; but here is a simple system that returns several solutions:
Yes you can! Like I said, if I type
Solve[M.{a,b,c}=={0,0,0},{a,b,c}]
I get
{{a -> 0. + 0.0410026 c, b -> 0. + 1.35294 c}}
Geometrically, this is a straight line in space which passes through the
origin. Such a line as *obviously* two points {a,b,c} with norm 1 (those
points are symmetric with respect to the origin, of course).
You can even do the computation by hand: if you want to get the points
{a,b,c} such that
1) a = 0.0410026 c;
2) b = 1.35294 c;
3) a^2 + b^2 + c^2 = 1
then these points are +/-{0.0243644,0.803937,0.594215}.
Best regards,
Jose Carlos Santos
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