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\title{Math 122: Homework 2}
\author{}
\date{}
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\begin{itemize}[noitemsep]
  \item Section 10.2 \# 10
  \item Section 10.4 \# 2,6,11
	\item Linear Algebra Problems 5,6,7
\end{itemize}

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{\textbf{Section 10.2 \#10}}. Let $R$ be a commutative ring. Prove that
$\operatorname{Hom}_R (R, R)$ and $R$ are isomorphic as rings.

\

{\textbf{Section 10.4 \#2.}} Show that the element ``$2 \otimes 1$'' is
zero in $\mathbb{Z} \otimes_{\mathbb{Z}} (\mathbb{Z}/2\mathbb{Z})$ but is
nonzero in $2\mathbb{Z} \otimes_{\mathbb{Z}} (\mathbb{Z}/ 2\mathbb{Z})$.

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{\textbf{Section 10.4 \#6}}. If $R$ is any integral domain with field of
fractions $Q$, prove that
\[ (Q / R) \otimes_R (Q / R) = 0. \]


{\textbf{Section 10.4 \#11.}} Let $\{ e_1, e_2 \}$ be a basis of $V
=\mathbb{R}^2$. Show that the element $e_1 \otimes e_2 + e_2 \otimes e_1$ in
$V \otimes_{\mathbb{R}} V$ cannot be written as a simple tensor $v \otimes w$
for any $v, w \in \mathbb{R}^2$.

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The following linear algebra problems are not from Dummit and Foote. If $V$ is a vector space
over a field $F$, then let $V^{\ast} = \operatorname{Hom}_F (V, F)$ be the {\textit{dual
space}}. If $W$ is another vector space over $V$ and $f : V \longrightarrow W$
is a linear map, define $f^{\ast} : W^{\ast} \longrightarrow V^{\ast}$ be
composition with $f$. This is the {\textit{dual map}}.

\

{\textbf{Linear Algebra Problem 5.}} Prove that if $f : V \longrightarrow W$ and $g : U
\longrightarrow V$ are linear transformations then $(f \circ g)^{\ast} =
g^{\ast} \circ f^{\ast}$.

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{\textbf{Linear Algebra Problem 6.}} Suppose that $U$ and $V$ are finite-dimensional
vector spaces. Define a linear map $\gamma : U^{\ast} \times V \longrightarrow
\operatorname{Hom}_F (U, V)$ by letting $\gamma (u^{\ast}, v) : U \mapsto V$ be the
map that sends $u \in U$ to $u^{\ast} (u) v$. Check that $\gamma$ is bilinear.
So by the universal property of the tensor product, this induces a map
\[ \delta : U^{\ast} \otimes V \longrightarrow \operatorname{Hom}_F (U, V) . \]
Prove that $\delta$ is a vector space isomorphism.

\

{\textbf{Linear Algebra Problem 7.}} Suppose now that $f : V \longrightarrow W$ is a
vector space homomorphism. Show that composition with $f$ is a linear map
$f_{\ast} : \operatorname{Hom}_F (U, V) \longrightarrow \operatorname{Hom} (U, W)$ and that
there is a commutative diagram (with $\delta$ as in Problem~6):
\[
\begin{tikzpicture}[scale=1.2]
\node at (0,0) {$U^\ast\otimes V$};
\node at (0,-2) {$U^\ast\otimes W$};
\node at (4,0) {$\operatorname{Hom}(U,V)$};
\node at (4,-2) {$\operatorname{Hom}(U,W)$};
\draw[->, >=Stealth, shorten >=30pt, shorten <=30pt] (0,0) -- (4,0) node[scale=.8,midway, above] {$\delta$};
\draw[->, >=Stealth, shorten >=30pt, shorten <=30pt] (0,-2) -- (4,-2) node[scale=.8,midway, above] {$\delta$};
\draw[->, >=Stealth, shorten >=10pt, shorten <=10pt] (0,0) -- (0,-2) node[scale=.8,midway, left] {$1_{U^\ast}\otimes f$};
\draw[->, >=Stealth, shorten >=10pt, shorten <=10pt] (4,0) -- (4,-2) node[scale=.8,midway, right] {$f_\ast$};
\end{tikzpicture}
\]
If $X$ is another vector space, can you do something similar with a linear map
$g : X \longrightarrow U$?

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