Homogenious Eqns: Solns Apart from Trivial Solution?

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SUMMARY

In the discussion regarding homogeneous equations, it is established that if the number of linearly independent homogeneous equations equals the number of variables, the only solution is the trivial solution where all variables are zero. This conclusion applies specifically to linear equations. For nonlinear equations, the solutions can vary significantly based on the specific equations involved, indicating a broader range of potential outcomes.

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  • Understanding of linear independence in equations
  • Familiarity with homogeneous equations
  • Basic knowledge of linear algebra concepts
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Students and professionals in mathematics, particularly those studying linear algebra, as well as researchers exploring the solutions of equations in various mathematical contexts.

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if we have same number of linearly independent homogenious equations as the number of variables , then how many solutions will we get apart from the trivial solution ?
 
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If you are talking about linear equations that are linearly indepdent, the only solution is the trivial one with all the variables equal to zero.

For nonlinear equations, anything might happen, depending on the particular equations.
 
thanks alephzero !
 

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