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April 27th, 2013, 11:41 AM  #1 
Newbie Joined: Apr 2013 Posts: 11 Thanks: 0  Transformation is not invertible
T is a linear transformation in (^3,^2) S is a linear transformation in (^2,^3) Prove that the transformation ST in (^3) is not invertible. Prove (or give example) that TS in (^2) can be invertible Thanks a lot in advance 
April 27th, 2013, 02:47 PM  #2 
Newbie Joined: Apr 2013 Posts: 10 Thanks: 0  Re: Transformation is not invertible
Note that the image of ST is contained in the image of S, and the image of S can be at most 2dimensional, i.e. . Hence, ST has no chance of mapping onto . On the other hand, if we define T and S by T(x,y,z) = (x,y) and S(x,y) = (x,y,0) (note these are both linear), then TS(x,y)=(x,y) is the identity map on , which is clearly invertible. 
April 27th, 2013, 03:42 PM  #3 
Newbie Joined: Apr 2013 Posts: 11 Thanks: 0  Re: Transformation is not invertible
Thanks a lot greentunic. I am just wondering about one property. It says if T is a linear transformation in (F^3,F^2) S is a linear transformation in (F^2,F^3) Then ST is a linear transformation in (F^3) Isn't it? Then ST should map onto F^3 Am I wrong? Thanks again. 
April 28th, 2013, 04:49 PM  #4 
Newbie Joined: Apr 2013 Posts: 10 Thanks: 0  Re: Transformation is not invertible
ST is a linear map from into , but it does not map onto . In other words, ST is not surjective.


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