Associated Homogeneous System definition

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SUMMARY

The "associated homogeneous system" refers to a system of linear equations where the right-hand side of each equation is set to zero. This system can be expressed in mathematical notation as follows: for a given system of equations represented by a_{ij}x_j = b_i, the associated homogeneous system is a_{ij}x_j = 0. This transformation is crucial in linear algebra as it allows for the analysis of solutions to the system, particularly in determining the existence of non-trivial solutions.

PREREQUISITES
  • Understanding of linear algebra concepts
  • Familiarity with systems of linear equations
  • Knowledge of mathematical notation for equations
  • Basic grasp of vector spaces and linear combinations
NEXT STEPS
  • Study the properties of homogeneous systems in linear algebra
  • Learn about the rank and nullity of matrices
  • Explore methods for solving systems of linear equations, such as Gaussian elimination
  • Investigate the implications of the homogeneous system on vector spaces
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Students and professionals in mathematics, particularly those studying linear algebra, as well as educators looking to explain the concept of homogeneous systems in a clear and structured manner.

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Hi, thank you for viewing this thread. I have been googling for its definition for quite a while, but have not found any yet. Just wondering if there is a definition of it, in mathematical notations and in words?
 
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Any system of linear equations can be written in the form
a_{11}x_1+ a_{12}x_2+ \cdot\cdot\cdot+ a_{1n}x_n= b_1
a_{21}x_1+ a_{22}x_2+ \cdot\cdot\cdot+ a_{2n}x_n}= b_2
etc.

That is, the left side of each equation is a linear combination of the variables, and the right side is a number. The "associated homogenous system" is exactly the same with the numbers on the right side all set to 0.

The "associated homogenous system" for the above system is:
a_{11}x_1+ a_{12}x_2+ \cdot\cdot\cdot+ a_{1n}x_n= 0
a_{21}x_1+ a_{22}x_2+ \cdot\cdot\cdot+ a_{2n}x_n= 0
etc.
 
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Alright, I got it. Thanks for your help!
 

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