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April 27th, 2013, 11:41 AM   #1
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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
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April 27th, 2013, 02:47 PM   #2
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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 2-dimensional,
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.
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April 27th, 2013, 03:42 PM   #3
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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.
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April 28th, 2013, 04:49 PM   #4
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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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