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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 ?
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.
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.
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.
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.
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.