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Not Invertible Linear Transformations

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This problem is #6 from pg. 83 of Hoffman & Kunze Linear algebra. A thorough explanation of the problem would be greatly appreciated.

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Not-Invertible Linear Transformations are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

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by way of example, consider T: R^3 --> R^2 to be T(a, b, c) = (a, b) and

U : R^2 --> R^3 defined by U(a, b) = (a, b, 0)

then UT is defined by UT(a, b, c) = (a, b, 0) which is not invertible (it's the projection onto the first two ...

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