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Math Help - [Matrix] Q about Trivial&NonTrivial System

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    [Matrix] Q about Trivial&NonTrivial System

    I just like to check

    Trivial solution = linearly independent
    = unique = when n x m matrix, n=m=rank

    NonTrivial solution = linearly independent
    = infinite = row of zeros = n>rank



    Also. (True of False)

    (a) If the system is homogeneous, every solution is trival.
    - False, since if the matrix contains a row of zeros (0 0 ... 0|0), then the system is non-trival.

    (b) If the system has a nontrivial solution, it cannot be homogeneous.
    - False, since homogeneous system is either unique(trival) or infinite(nontrival). Therefore, nontrivial solution must be a homogenous system.

    (c) If there exists a trivial solution, the system is homogeneous.
    - True, since homogenous system is either unique(trival) or infinite(nontrival). Therefore, trivial solution must be a homogeneous system.

    (d) If the system is consistent, it must be homogenous.
    - True, since any system that is not inconsistent is called consistent and consistent either has unique or infinite solutions which is a property of a homogeneous system.

    Now assume that the system is homogeneous:

    (e) If there exists a nontrivial solution, there is no trivial solution.
    - True. A matrix with a nontrivial solution must contain a row of all zeros but no trivial solution must not contain a row of zeros in RREF.(What is no trival soln?)

    (f) If there exists a solution, there are infinitely many solutions.
    - False. If the system has basic solutions (systems without a row of zeros), the system contains unique solutions.

    (g) If there exist nontrivial solutions, the row-echelon form of A has a row of zeros.
    - True, since linearly dependent systems requires a row of zeros for parameters, which results infinite solutions.

    (h) If the row-echelon form of A has a row of zeros, there exist nontrival solutions.
    ((((((((((((((((((((I think it is possible. I believe what my Professor said... ??? but y??)))))))))))))))))))

    (i) If a row operation is applied to the system, the new system is also homogeneous.
    -True. If the starting matrix is a homogenous system then the applied row operation system is always a homogenous system because it does not change property of original matrix.

    SORRY, IT IS BIT LONG...
    I LIKE TO KNOW THE ANSWERS ARE CORRECT AND IF IT IS WRONG CAN YOU PLEASE FIX IT IN DETAIL. IF YOU CAN ADD MORE INFO., PLEASE DO.
    THANKS.
    Last edited by UjNique88; October 14th 2009 at 09:14 AM.
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