hw5sol - HOMEWORK 5 - SOLUTIONS For all of the following...

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HOMEWORK 5 - SOLUTIONS For all of the following exercises, decide if the statement is true or false, and then prove your decision correct. 1. False. We know that two vector spaces are isomorphic iﬀ they have the same dimension, but this doesn’t mean that any linear transformation between isomorphic vector spaces must be an isomorphism. In this case, T could’ve just been the zero map. 2. (GRADED - EASY). False. This is almost true, what’s missing is the hypothesis that V and W have the same ﬁnite dimension. For instance if T : R → { 0 } is a (in fact the only) linear transformation from R to the trivial vector space consisting of just a zero vector, then T is surjective but not one-to-one. 3. False. We proved in class that a linear transformation is injective iﬀ its nullity is 0, and the statement/proof of this fact had nothing to do with whether or not the domain/codomain were ﬁnite-dimensional. 4. (GRADED - MEDIUM) True. By the Extending by Linearity Theorem, a linear

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This note was uploaded on 03/28/2012 for the course MATH 110 taught by Professor Gurevitch during the Fall '08 term at University of California, Berkeley.

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hw5sol - HOMEWORK 5 - SOLUTIONS For all of the following...

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