What is the name of this principle?

[natski]

Hi all,

When solving a set of equations with n unknown parameters, you need at least n equations to do this, but perhaps more. What is the name of this principle?

Cheers,
Natski

 

[HallsofIvy]

The "principle" you state is not true. For example, the equation x²+ y²+ z²= 0, although only one equation in 3 unknown parameters has the unique solution x= y= z= 0 in the real numbers. If that is not what you meant, please explain more.

 

[natski]

Ok, but you can't get all of the solutions with just that one equation. Perhaps a better problem would be to consider n non-linear equations.

 

[yuiop]

They are sometimes known as simultaneous equations.

 

[natski]

Yup, I know that. I want to know the name of principle which dictates how many equations you need in order to find the solutions to n parameters.

 

[nasu]

The "principle" you state is not true. For example, the equation x²+ y²+ z²= 0, although only one equation in 3 unknown parameters has the unique solution x= y= z= 0 in the real numbers. If that is not what you meant, please explain more.

Who said the solutions must be real?
1,1 and i*sqr(2) is also a solutions. And many other sets.

 

[nasu]

Ok, but you can't get all of the solutions with just that one equation. Perhaps a better problem would be to consider n non-linear equations.

The "principle" refers usually to linear equations. And n equations must be linear independent, if you want to get n unknowns.
I don't think I ever seen this called a principle and even less given some specific name.

 

[Bill_B]

According to http://planetmath.org/encyclopedia/RankNullityTheorem.html, the rank-nullity theorem says that "the number of variables minus the number of independent linear constraints equals the number of linearly independent solutions."

I don't think this is exactly what you're asking for, but it's as close as I could get.

 

[Office_Shredder]

Who said the solutions must be real?
1,1 and i*sqr(2) is also a solutions. And many other sets.

Well hell, who says the solutions must be complex? (1,1,1) is a solution in Z₃

 

[HallsofIvy]

Who said the solutions must be real?
1,1 and i*sqr(2) is also a solutions. And many other sets.

I did. And since the original question just referred to 'a system of equations' without any restrictions, I am free to state a problem dealing with any system of equations in whatever number system I choose as a counter-example.

 

[HallsofIvy]

Ok, but you can't get all of the solutions with just that one equation. Perhaps a better problem would be to consider n non-linear equations.

Are you suggesting that there are solutions to x²+ y²+ z²= 0 other than x= y= z= 0? If so, tell me what they are! If not, I just gave you an example of 1 non-linear equation which, by itself, determines 3 solutions.

In fact, in the real number system, given any positive integer n, the system
$$x_1^2+ x_2^2+ \cdot\cdot\cdot+ x_n^2= 0$$
completely determines all n solutions. What you are attempting to assert is true of linear systems of equations, not systems of equations in general.

 

[CRGreathouse]

What you are attempting to assert is true of linear systems of equations, not systems of equations in general.

Halls, what (if anything!) can be said about other systems of equations (perhaps polynomials of degree < d)?

 

[Office_Shredder]

Are you suggesting that there are solutions to x²+ y²+ z²= 1 other than x= y= z= 0? If so, tell me what they are!

I'm tempted to type an uncountable list of solutions here, but I probably would be at it all night

 

[HallsofIvy]

I'm tempted to type an uncountable list of solutions here, but I probably would be at it all night

And I am tempted to call myself an uncountable list of names! Of course, I meant x²+ y²+ z²= 0, as I had initially.

The point is still that the "principle" enuciated in the original post simply does not exist!

 

[maze]

I'm tempted to type an uncountable list of solutions here, but I probably would be at it all night

If you are thinking of complex x,y,z, then the example can be easily extended to

$$x \bar{x} + y \bar{y} + z \bar{z} = 0$$

so that there is only 1 solution.

 

[HallsofIvy]

No, he was thinking of my unfortunate typo where I wrote "= 1" rather than "= 0" but thank you for extending my point to the complex numbers.

There is no "name" for the principle initially enunciated because we are not in the practice of giving names to incorrect statements!

 

[Werg22]

I think the OP is referring to systems of linear equations. In that case, it's not a principle but simply a trivial fact easily seen when studying systems of a linear equations in their matrix representations: if the number of variables exceeds the number of equations, there must be free variables and hence an infinite number of solutions. In general, there exists no more than 1 solution when there are n linearly independent equations in F^n, where F is the field from which the entries of the matrix (coefficients of the equations) are coming.

 

[arildno]

if the number of variables exceeds the number of equations, there must be free variables and hence an infinite number of solutions.

Totally incorrect.

For example, the equation:

x+y+z=x+y+z+1 has NO solutions, not infinitely many.

 

[HallsofIvy]

The original poster said specifically, in post 3, "Perhaps a better problem would be to consider n non-linear equations. "

 

[Werg22]

Totally incorrect.

For example, the equation:

x+y+z=x+y+z+1 has NO solutions, not infinitely many.

Why do you have to be so obnoxious? Obviously I did not mean this applied to systems with inconsistent equations. There are more civil ways to correct someone.

The original poster said specifically, in post 3, "Perhaps a better problem would be to consider n non-linear equations. "

Sorry about that.

 

[arildno]

Why do you have to be so obnoxious? Obviously I did not mean this applied to systems with inconsistent equations. There are more civil ways to correct someone.

Wherein lies my "obnoxity"? What you actually wrote WAS "totally incorrect".


As for the obviosity of that crucial condition you NOW say you place upon what you wrote, you cannot expect that I, or anybody else, can look inside your head to find all your implicit assumptions, however obvious they might be to you.