Homework #1 for Math H110, fall '02
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1. A couple of people correctly pointed out that the proof of zero mapping
to zero in class wasn't complete. There are a number of ways to fix it
(by assuming additional axioms), but instead let's construct a
pathological example.
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Let F_2 be the field with two elements {0,1}, so 1+1=0. Find 
a "vector space" V over F_2, satisfying the axioms given in class:
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<li>vector addition is commutative associative, with an additive identity
<li>(e+f)v = ev + fv
<li>(ef)v = e(fv)
<li>1 times v = v for all v
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but with three elements (what would its dimension be, log_2 of 3?).

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Hint: convince yourself that if we assume that V has not only + but -,
this couldn't happen. So V's weird + can't have cancelation
(a+b = a+c implying b=c).
Also, if we assumed that 0 times v always gives the same vector,
we could fix it. So your V will have to break that too. Yuck!

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(Definitely the preferred way is to assume that V has + and -, so that
on its own without F it's an object worthy of attention in Math 113.)

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2. Curtis p15 #6.
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3. Curtis 2.4 #5,9,10
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4. Let V be a fin. gen. vector space, and X,Y two subspaces of the
same dimension. 
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<li>Show that there is an invertible map T : V->V such that T(X) = Y.
<li>Show that there is a map T: V->V such that T(X) = T(V) = Y.
When is it invertible?
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