1.3 Homogeneous Equations

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# 1.3 Homogeneous Equations - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about '1.3 Homogeneous Equations' - yetta-terrell

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Presentation Transcript
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
• 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
• 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. 