How to decide if T maps R^5 onto R^5
 
  
 
  
 
 
 
 
 
 
    
        
        asked 7 years ago  
 
 beartato327 gravatar image  
 
 
 
 
    
        
        updated 7 years ago  
 
 FrédéricC gravatar image  
 
 
 
 
 
Hi all,
 
New to SageMath, for a homework problem I have to use SageMath to solve a problem and this problem is, "Decide if T maps R^5 onto R^5." This is the matrix ([[9,13,5,6,-1],[14,15,-7,-6,4],[-8,-9,12,-5,-9],[-5,-6,-8,9,8],[13,14,15,2,11]]). How do I go about proving this in SageMath?
 
Thanks
 
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This is not a question about Sage, but about the mathematics you learnt. Which notion is useful to know whether a linear map in onto ?
tmonteil gravatar imagetmonteil (
    
        7 years ago
    )
I decided on a similar site to give the code in similar situations, thus offering a change to understand math supported by computer soft and computer examples.
(I will work over rationals, well −⊗QR is exact.)
Which of the following computations does (not) answer the problem?
sage: A = matrix( QQ, [ [ 9,13, 5, 6,-1] ,
....:                   [14,15,-7,-6, 4] ,
....:                   [-8,-9,12,-5,-9] ,
....:                   [-5,-6,-8, 9, 8] ,
....:                   [13,14,15, 2,11] ] )

sage: A.det()
0

sage: A.rank()
4

sage: ImA = A.image()
sage: ImA.dimension()
4

sage: KerA = A.kernel()
sage: KerA.dimension()
1
dan_fulea gravatar imagedan_fulea (
    
        7 years ago
    )
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