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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 onetoone.
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 ﬁnitedimensional.
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.
 Fall '08
 GUREVITCH
 Math, Vector Space

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