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Does a homogeneous system of 4 equations in 5 variables have infinitely many...

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Does a homogeneous system of 4 equations in 5 variables have infinitely many solutions????

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Posted (Answer #1)

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Yes!

 Because the matrix corresponding to coefficients of the variables has its rank less than the number of variables. In such a case at least one variable will be free variable and hence can take infinitely many values from corresponding field.

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Posted (Answer #2)

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Homogeneous systems of linear equations are systems where the constant term in each equations is zero. Therefore, they have the form [A|0], where A is the matrix of coefficients of the variables in the system of equations. Systems of this type always have a solution. There is always the trivial solution where [x1, x2, ..., xn] = [0,0,...0].

Apart from this ,we four equation in 5 variables, which means we have one free variables and we can assign any arbitrary value to it . This implies we have an infinite no. of solutions.

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