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Elzair
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Proving a linear system of equations cannot have more than one finite solutions
Prove that the number of solutions to a linear system can not be a finite number larger than one. Provide either a general proof or for a system with two equations and two unknowns.
It is mostly simple algebra.
Assume the following:
[tex]a_{11}x_{1}+a_{12}x_{2}=b_{1}[/tex] NOTE: If the tex messed up, this reads a11 * x1 + a12 * x2 = b1
[tex]a_{21}x_{1}+a_{22}x_{2}=b_{2}[/tex] and a21 * x1 + a22 * x2 = b2
There exists two unique solution sets to the system: {[tex]c_{1}[/tex], [tex]c_{2}[/tex]} (NOTE: this reads {c1, c2}) and {[tex]d_{1}[/tex], [tex]d_{2}[/tex]} are solutions, and [tex]c_{1} \neq d_{1}[/tex], [tex]c_{2} \neq d_{2}[/tex].
.:.
[tex]a_{11}c_{1}+a_{12}c_2}=b_{1}=a_{11}d_{1}+a_{12}d_2}[/tex]
[tex]a_{21}c_{1}+a_{22}c_2}=b_{2}=a_{21}d_{1}+a_{22}d_2}[/tex]
=>
[tex]a_{11} \left( c_{1}-d_{1} \right) + a_{12} \left( c_{2}-d_{2} \right) =0[/tex]
[tex]a_{21} \left( c_{1}-d_{1} \right) + a_{22} \left( c_{2}-d_{2} \right) =0[/tex]
Therefore, the system is now homogenous. It should be easy enough to generalize to an nxn system of equations. Also, I could generalize it to x number of solutions by showing that any two of the potential solutions subtracted by each other would yield a homogenous system.
Suppose [tex]f_{1}=c_{1}-d_{1}[/tex], [tex]f_{2}=c_{2}-d_{2}[/tex]
.:.
[tex]a_{11}f_{1}+a_{12}f_{2}=0[/tex]
[tex]a_{21}f_{2}+a_{22}f_{2}=0[/tex]
[tex]f_{1}=-\frac{a_{12}}{a_{11}}f_{2}[/tex]
[tex]a_{21} \left( \frac{-a_{12}}{a_{11}} f_{2} \right) + a_{22}f_{2} = \left( a_{22} - \frac{a_{12}a_{21}}{a_{11}} \right) f_{2} = 0[/tex]
Now, there are two possibilities:
1.) [tex]f_{2}=f_{1}=0[/tex] ergo [tex]c_{1}=d_{1}[/tex] and [tex]c_{2}=d_{2}[/tex], which is a CONTRADICTION
2.) [tex]a_{11}a_{22} - a_{21}a_{12} = 0[/tex] ergo det(a)=0
Now, how should I prove that a homogeneous linear system of equations whose coefficient matrix has a determinant of zero has infinite solutions?
Furthermore, how should I go about generalizing the problem to an arbitrary number of equations, unknowns and solutions? It should be easy to show that the difference of any two of a given x solutions to an nxn linear system of equations results in a homogeneous system. However, I am unsure of how to state that symbolically. Furthermore, I am a little unclear on how I should generalize the proof that the determinant of the coefficient matrix must be zero.
Homework Statement
Prove that the number of solutions to a linear system can not be a finite number larger than one. Provide either a general proof or for a system with two equations and two unknowns.
Homework Equations
It is mostly simple algebra.
The Attempt at a Solution
Assume the following:
[tex]a_{11}x_{1}+a_{12}x_{2}=b_{1}[/tex] NOTE: If the tex messed up, this reads a11 * x1 + a12 * x2 = b1
[tex]a_{21}x_{1}+a_{22}x_{2}=b_{2}[/tex] and a21 * x1 + a22 * x2 = b2
There exists two unique solution sets to the system: {[tex]c_{1}[/tex], [tex]c_{2}[/tex]} (NOTE: this reads {c1, c2}) and {[tex]d_{1}[/tex], [tex]d_{2}[/tex]} are solutions, and [tex]c_{1} \neq d_{1}[/tex], [tex]c_{2} \neq d_{2}[/tex].
.:.
[tex]a_{11}c_{1}+a_{12}c_2}=b_{1}=a_{11}d_{1}+a_{12}d_2}[/tex]
[tex]a_{21}c_{1}+a_{22}c_2}=b_{2}=a_{21}d_{1}+a_{22}d_2}[/tex]
=>
[tex]a_{11} \left( c_{1}-d_{1} \right) + a_{12} \left( c_{2}-d_{2} \right) =0[/tex]
[tex]a_{21} \left( c_{1}-d_{1} \right) + a_{22} \left( c_{2}-d_{2} \right) =0[/tex]
Therefore, the system is now homogenous. It should be easy enough to generalize to an nxn system of equations. Also, I could generalize it to x number of solutions by showing that any two of the potential solutions subtracted by each other would yield a homogenous system.
Suppose [tex]f_{1}=c_{1}-d_{1}[/tex], [tex]f_{2}=c_{2}-d_{2}[/tex]
.:.
[tex]a_{11}f_{1}+a_{12}f_{2}=0[/tex]
[tex]a_{21}f_{2}+a_{22}f_{2}=0[/tex]
[tex]f_{1}=-\frac{a_{12}}{a_{11}}f_{2}[/tex]
[tex]a_{21} \left( \frac{-a_{12}}{a_{11}} f_{2} \right) + a_{22}f_{2} = \left( a_{22} - \frac{a_{12}a_{21}}{a_{11}} \right) f_{2} = 0[/tex]
Now, there are two possibilities:
1.) [tex]f_{2}=f_{1}=0[/tex] ergo [tex]c_{1}=d_{1}[/tex] and [tex]c_{2}=d_{2}[/tex], which is a CONTRADICTION
2.) [tex]a_{11}a_{22} - a_{21}a_{12} = 0[/tex] ergo det(a)=0
Now, how should I prove that a homogeneous linear system of equations whose coefficient matrix has a determinant of zero has infinite solutions?
Furthermore, how should I go about generalizing the problem to an arbitrary number of equations, unknowns and solutions? It should be easy to show that the difference of any two of a given x solutions to an nxn linear system of equations results in a homogeneous system. However, I am unsure of how to state that symbolically. Furthermore, I am a little unclear on how I should generalize the proof that the determinant of the coefficient matrix must be zero.