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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

1851.0. "Linear Independence" by PASTA::SUDHARSANAN () Fri Mar 11 1994 19:10

    
    I have the following question on linear independence.  
    
    Let f(.) be a function which is strictly monotonic, continuous, and 
    a first and third quadrant non-linearity, i.e. x f(x) > 0 for 
    all x != 0  and f(x) = 0 for x =0.  Also 
               
                 |f(x)|       |f(y)|
                 ------   <=   ------      for all |x| >= |y|  
                  |x|          |y|                     
    
    Now, let the columns of a matrix X with elements x_{i,j} be linearly 
    independent.  Will the columns of the matrix F with elements 
    f(x_{i,j}) be linearly independent?   
    
    Curiously,
                    
    S.  Sudharsanan
    
    pasta::sudharsanan
T.RTitleUserPersonal
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1851.1CounterexampleWIBBIN::NOYCEDEC 21064-200DX5 : 130 SPECint @ $36KMon Mar 14 1994 13:393
               [ 3   4 ]                   [ 3  4 ]
No.  Given X = [       ] we could have F = [      ]
               [ 6  10 ]                   [ 6  8 ]
1851.2More Restrictions on the functionPASTA::SUDHARSANANMon Mar 14 1994 15:5118
    re.1
    
    I can actually further restrict my function f(.) as follows,
    
                |f(x)|             |f(y)|
                ------      <      ------
                 |x|                |y|        
    
       for all   |x| >  |y|.
    
    Also, I can impose the condition that f(.) be a C-infinity function,
    i.e. it is continuously differentiable infinitely many times.  
    
    The above restrictions will make the counterexample invalid, however
    the claim in the original note may still not be true. 
    
    S. Sudharsanan