How to decide if T maps R^5 onto R^5

asked 2017-06-02 11:24:59 -0500

updated 2017-07-31 14:14:48 -0500

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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Comments

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 ( 2017-06-02 17:45:31 -0500 )edit

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 $-\otimes_{\mathbb Q}\mathbb R$ 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 ( 2017-06-04 06:47:41 -0500 )edit