How do you know if a system of N linear equations with N unknowns, has N unique solutions?
Linear equations help
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linear-algebra
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1Solutions are unique if and only if the matrix has nonzero determinant. – 2012-11-28
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3And if the solution is unique, there is only $1$ solution. This solution may consist of specifying the values of $N$ unknowns. – 2012-11-28
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0By 1 solution do you mean the values of all N unknowns? – 2012-11-28
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0Im confused, when one says a system of n equations with n unknowns has 1 unique solution, does that mean their is a solution for all n unknown values? – 2012-11-28
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0Ethan, For example, the system of equations $\{x+y=0,x-y=0\}$ has the unique solution $(x,y)=(0,0)$ (if the characteristic is not $2$). – 2012-11-28
1 Answers
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Okay, think of a system of two linear equations in two unknowns, $x$, $y$. If a unique solution exists, then there exist a unique $m, n$ such that $x = m, \; y = n$. That is the unique solution. This solution might be expressed as the ordered pair $(m, n)$.
For any system of $n$ linear equations in $n$ unknowns, one and only one of the following is true:
The system of linear equations has:
- An infinite number of solutions.
- No solution.
- A unique solution.
If more than one solution exists, then an infinite number of solutions exist.
If no solution exists, then the linear system is "inconsistent."
A unique solution exists if and only if the determinant of the coefficient matrix is nonzero.
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0I know what a linear system of equations is, what I am now asking is, that if one says a system has a "unique solution", does that mean there is a value for each unknown that satisfies each equation. – 2012-11-28
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0Yes, that means that each unknown has exactly one value that satisfies all the equations. If the unknowns are x, y, z, then there is one value for x, and one value for y, and one value for z that satisfy all the linear equations. It is ***not*** necessary that x = y = z. – 2012-11-28
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0Does that clear things up, Ethan? – 2012-11-28
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0Yes thanks alot – 2012-11-28