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Question
I am having difficulty figuring out if these statements are true or false. Any help would be greatly appreciated.

Determine whether the statement is true or false.
(a)   A homogeneous system may be inconsistent.
(b)   A homogeneous linear system has infinitely many solutions if and only if it has free variables.
(c)   If all variables of a homogeneous linear system are leading, then the system has only the trivial solution.

Answer
a) False: Having all the variables = 0 is always a solution (called the trivial solution).

b) True: Free variables can take on any value thus generating infinitely many solutions.

c) If "leading" is the same as "basic", meaning the opposite of free, then the trivial solution is the only solution.

Randy

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randy patton

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college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

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26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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