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1.3 Homogeneous Equations. Definitions. A system of equations is homogeneous if the constant term in each equation is 0. The form of each equation will be the following:.

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definitions
Definitions
  • A system of equations is homogeneous if the constant term in each equation is 0. The form of each equation will be the following:
  • One solution would be for all of the variables, x1, x2,..., xn, to be 0. This is the trivial solution.
  • Any solution which allows any variable to be non-zero is non-trivial.
example
Example
  • Determine whether the following homogeneous system will have any non-trivial solutions.
  • The existence of parameters in the solution guaranteed that there would be a non-trivial solution (in fact an infinite number of non-trivial solutions).
  • Theorem: If a homogeneous system of linear equations has more variables than equations, then it has infinitely many non-trivial solutions.
proof
Proof
  • Theorem: If a homogeneous system of linear equations has more variables than equations, then it has infinitely many non-trivial solutions.
  • Proof:
    • Take a system of m homogeneous equations in n variables where m < n.
    • We always have at least the trivial solution (so not inconsistent)
    • In reduced row-echelon form, this system will have r leading variables, and n - r free variables, and therefore n - r parameters in the solution.
    • As long as there is at least 1 parameter, we know we have infinitely many non-trivial solutions.
    • Therefore, all we need to show is that n - r > 0 or n > r.
    • In a system of m equations, the greatest number of leading variables is m, so r ≤ m.
    • We know that m < n, so r < n.