Homogenious Eqns: Solns Apart from Trivial Solution?

  • Thread starter phymatter
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In summary, a homogenous equation is a mathematical equation where all the terms have the same degree and are composed of the same variables. It can be solved by finding the eigenvalues and eigenvectors of the corresponding matrix and writing the general solution as a linear combination of the eigenvectors. A trivial solution to a homogenous equation is when all variables are equal to zero, and it is possible for a homogenous equation to have multiple non-trivial solutions. Homogenous equations are important in science, as they can be used to model and analyze physical systems, and are a fundamental concept in linear algebra and other areas of mathematics.
  • #1
phymatter
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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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  • #2
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.
 
  • #3
thanks alephzero !
 

1. What is a homogenous equation?

A homogenous equation is a mathematical equation where all the terms have the same degree and are composed of the same variables. This means that the equation can be written in the form of Ax = 0, where A is a matrix and x is a vector.

2. What is a trivial solution?

A trivial solution is a solution to a homogenous equation that satisfies the equation by making all variables equal to zero. In other words, it is a solution where all the unknowns are equal to zero.

3. How do you solve a homogenous equation?

To solve a homogenous equation, you need to find the eigenvalues and eigenvectors of the corresponding matrix A. Then, the general solution can be written as a linear combination of the eigenvectors multiplied by a constant. The values of the constants can be determined by using the initial conditions of the problem.

4. Can a homogenous equation have multiple non-trivial solutions?

Yes, a homogenous equation can have multiple non-trivial solutions. This is because the eigenvectors of the matrix A can form a basis for the solution space, and any linear combination of these eigenvectors will also be a solution.

5. What is the importance of homogenous equations in science?

Homogenous equations are important in science as they often arise in physical systems and can be used to model and analyze a variety of phenomena. They are also a fundamental concept in linear algebra and are used in many areas of mathematics, such as differential equations and optimization problems.

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