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{\Large 18.099 - 18.06CI.}
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{\large HW-4}
{Due on Monday, March 15 in class.
First draft due by Thursday, March 11.}
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\begin{enumerate}
\item{ \begin{enumerate}
\item{A \emph{direct complement} to a subspace $L_1$ in a finite
dimensional space $L$ is a subspace $L_2 \subset L$ such that
$L = L_1 \oplus L_2$.
Prove that for any subspace $L_1$ a direct complement exists,
and the dimensions
of any two direct complements of $L_1$ in $L$ coincide. }
\item{For a linear map $F : L \to M$, let ${\rm coker}F$ be a direct
complement of ${\rm Im}F$ in $M$. Define the \emph{index} of the map $F$ by
$$ {\rm ind}F = \dim ({\rm coker} F) - \dim ({\rm ker} F). $$
Check using (a) that ${\rm ind}F$ is well-defined.
Prove that if $L$ and $M$ are finite dimensional, ${\rm ind}F$
depends only on the dimensions of $L$ and $M$:
$$ {\rm ind}F = \dim (M) - \dim (L).$$ }
\item{Set $\dim(M) = \dim(L)=n$. What can you deduce from (b)
about systems of $n$ linear equations in $n$ variables?}
\end{enumerate} }
\item{Two ordered $n$-tuples of subspaces
$\langle L_1, L_2, \ldots , L_n\rangle$ and
$\langle L_1', L_2', \ldots L_n'\rangle$ in a finite dimensional $L$
are \emph{identically arranged} if there exists a linear automorphism
(bijecitve linear map from a space to itself) $F: L \to L$
such that $F(L_i)=L_i'$ for all $i =1, \ldots n$.
Show that all triples of non-coplanar, pairwise distinct lines through zero in
$\R^3$ are identically arranged. Classify the arrangements
of quadruples of such lines in $\R^3$. }
\end{enumerate}
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