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\def\title{Homework 4, February 26}
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\bs
These problems are due Monday, March 5.

Do p.~377, problems 1, 2, (3), (5), 6, 7, and p.~111, problems 1, 6, 7.

The exercises in parentheses are to be turned in only if you are uncertain
whether you got the correct answer.

\bs
The following exam question is to be done independently.
\bs

{\bf Exam question 1:} 
Suppose $F$ is a field, and $V$ is a finite dimensional $F$-vector space.
Suppose $U_1 \subseteq U_2 \subseteq \dots \subseteq U_n
$ are $F$-subspaces of $V$.  Let $U_1 \wedge U_2
\wedge \dots \wedge U_n $ denote the image of $U_1 \otimes_F U_2 \otimes_F
\cdots \otimes_F U_n \to \bigwedge^n V$.  If $X$ is a subspace
satisfying $U_p \subseteq X \subseteq U_{p+1}$, then show that there
is a natural isomorphism of
$$
{
U_1 \wedge \dots \wedge U_p \wedge U_{p+1} \wedge U_{p+2} \wedge\dots\wedge U_n
\over
U_1 \wedge \dots \wedge U_p \wedge X       \wedge U_{p+2} \wedge\dots\wedge U_n
}
$$
with
$$
\left(
U_1 \wedge \dots \wedge U_p
\right)
\otimes_F
\left(
{U_{p+1} \over X} \wedge {U_{p+2} \over X} \wedge\dots\wedge {U_n \over X}
\right)
$$
You might begin by trying to find a convenient set of generators and relations
for $U_1 \wedge \dots \wedge U_n$.  Notice that since you are asked to find a
natural isomorphism, it would be a bad idea for the isomorphism's definition to
depend on choices of basis.

\bye


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