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Re: Strange empty set of solutions


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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