---
date: [[2022-05-02]]
tags: #linear_algebra
cards-deck: default_obsidian
---
#math/linear_algebra
## system of linear equations

### definition
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a collection of equations that we try to solve simultaneously
![[Pasted image 20220502144937.png]]
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### equivalent if
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Two systems are equivalent if they have the same solution set. All solutions have to be the same.
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### theorem
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A system can either have:
- one solution
- no solution
- $\infty$ solutions
![[Pasted image 20220502145609.png]]
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[[matrix representation of systems of linear equations]]

### is it consistent?
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a system with one or more solutions is consistent. If it has no solution it's inconsistent. 
[[homogeneous]] systems are all consistent, since the zero solution (trivial solution) is always an answer. 
in the [[matrix representation of systems of linear equations#reduced row echelon form|reduced row echelon form]], a no solution matrix will have zeroes in the bottom row, except the last column. Basically saying, "0=1".
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