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homogeneous set of linear equations


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A set of m linear equations in n unknowns x1, x2,…, xn that has the form

a11x1+a12x2+⋯+a1nxn=0,

a21x1+a22x2+⋯+a2nxn=0,

am1x1+am2x2+⋯+amnxn=0.

Here, unlike the non-homogeneous case, the numbers on the right-hand sides of the equations are all zero. In matrix notation, this set of equations can be written Ax=0, where the unknowns form a column matrix x. Thus A is the m×n matrix [aij], andIf x is a solution of a homogeneous set of linear equations, then so is any scalar multiple kx of it. There is always the trivial solution x=0. What is generally of concern is whether it has other solutions besides this one. For a homogeneous set consisting of the same number of equations as unknowns, the matrix of coefficients A is a square matrix, and the set of equations has non-trivial solutions if and only if detA=0.

Subjects: Mathematics.


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