If I have a system of N linear equations with N unknowns, how do I know if each unknown has atleast one solution. I havn't taken any formal classes in linear algebra, so please if you use any abstract terminology associated with linear algebra, explain it please. I would appreciate any help.
Systems of equations
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linear-algebra
abstract-algebra
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0This is a useful method: http://en.wikipedia.org/wiki/Gaussian_elimination I don't think there is any simple way of going around linear algebra on a question like this (assuming that your N equations are of linear form). – 2012-11-27
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0I apologise I should have reworded my question better. I don't need help solving a system of linear equations, I want to know under what condinitions a system with n unknowns and n equations IS solveable. I am also not asking you to not use linear algebra, I am just asking you to explain any abstract notation, if you chose to use it. – 2012-11-27
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0Are we to assume these are linear equations? I think in general knowing in advance whether we can solve any set of $N$ equations in $N$ unknowns is very difficult, if not impossible. However if they are linear equations then there is a very simple test, calculate the determinant (http://en.wikipedia.org/wiki/Determinant). If it is non-zero (which is extremely likely) then you have a unique solution. If it is $0$ then you either have no solutions or infinitely many, and it is easy to check which is the case. – 2012-11-27
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0Yes all the equations are linear, and I don't even need unique solutions, I just need to know that all N unknowns have atleast one solution. Also I don't know what a "determinant" is, if you could help me out, id appreciate it. – 2012-11-27
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0no you can't always find solution, the coefficients must satisfy certain conditions – 2012-11-27
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0What conditions must they satisfy? – 2012-11-27
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0this is where matrices comes in. if one forms a matrix with the coefficients, the rows and columns must be linearly independent. – 2012-11-27
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0Im still fuzzy with linear algebra terminology, but when you say that two row/column vectors are linearly independent. That means none of their compoenents are scaled up versions of each other, right? – 2012-11-27
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0you're absolutely right – 2012-11-27
1 Answers
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- Take a look at this link explaining where a certain matrix is proven to be non invertivible using a system of linear equations.
- Also read this Wikipedia article explaining the properties of a system of linear equations.