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\pagestyle{fancy} \lhead{{\sf Homework 6 $|$ Due February 20
    (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Spring 2026 $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-2mm}


\begin{enumerate}

  %%------------------------------------------------------------------------

\item Let $U$, $V$, and $X$ be vector spaces over a field
  $K$.
  \begin{enumerate}
  \item Prove that the map $\tau\colon U\times V\longrightarrow U\otimes V$, defined by $\tau(u,v)=u\otimes v$, is bilinear.
  \item Prove that for any linear map $L\colon U\otimes V\to X$, the mapping
    $\beta:=L\circ\tau$ is bilinear.% from $U\times V$ to $X$.
  \item Prove the universal property of tensor products: for any
    bilinear map $\beta\colon U\times V\to X$, there is a unique
    linear mapping $L\colon U\otimes V\to X$ such that
    $\beta=L\circ\tau$:
  \[
  \begin{tikzpicture}[scale=3.5,>= stealth,
      every label/.style={font=\footnotesize}]
    \begin{scope}[shift={(0,0)}]
      \node (UxV) at (0,0) {$U\times V$};
      \node (X) at (1,0) {$X$};
      \node (UtensorV) at (.5,-.5) {$U\oplus V$};
      \draw[->] (UxV) -- (X) node[midway,above]{$\beta$};
      \draw[->] (UxV) -- (UtensorV) node[midway,below left]{$\tau$};
      \draw[->,dashed] (UtensorV) -- (X) node[midway,below right]{$L$};
    \end{scope}
  \end{tikzpicture}
  \]
    
  \end{enumerate}
  
  %%------------------------------------------------------------------------ 
    
\item If $\{u_1,\dots,u_n\}$ and $\{v_1,\dots,v_m\}$ are bases for $U$
  and $V$, then the pure tensors $\{u_i\otimes v_j\mid 1\leq i\leq
  n,\;1\leq j\leq m\}$ clearly span $U\otimes V$. Show that these are
  linearly independent, and conclude that $\dim(U\otimes V)=(\dim
  U)(\dim V)$. [\emph{Hint}: Apply the universal property to the
    canonical basis $\{f_{ij}\}$ of bilinear functions $U\times V\to
    K$.] \medskip
      
  %%------------------------------------------------------------------------ 
  
\item Prove $U\otimes V\cong V\otimes U$ by using the universal property of the tensor product. [\emph{Hint}: let $X=V\otimes U$.]
  
  %%------------------------------------------------------------------------ 

\item Use the universal property of the tensor product to prove 
  \[
  (U\otimes V)\otimes W\cong U\otimes(V\otimes W).
  \]

  %%------------------------------------------------------------------------ 

\item Use the universal property of the tensor product to prove 
  \[
  (U\times V)\otimes W\cong (U\otimes W)\times (V\otimes W).
  \]

%%------------------------------------------------------------------------ 

\item Let $W$ be a vector space with basis $\big\{w_{ij}\mid 1\leq
  i\leq n,\;1\leq j\leq m\big\}$. Define the linear map
  \[
  \alpha\colon W\longto U\otimes V,\qquad \alpha\colon w_{ij}\longmapsto u_i\otimes v_j.
  \]
  To show this is an isomorphism, we would like to define the (inverse) map
  $\beta\colon U\otimes V\to W$. But to do so, we would need a basis for
  $U\otimes V$. So instead, define a map
  \[
  \tilde\beta\colon F_{U\times V}\longto W,\qquad\tilde\beta\colon
  \text{\large $e$}_{\Sigma a_iu_i,\Sigma b_jv_j}\longmapsto
  \sum_{i,j} a_ib_jw_{ij}.
  \]
  where $F_{U\times V}$ is the \emph{free vector space} with basis $U\times V$:
  \[
  F_{U\times V}=\left\{\sum c_{uv}e_{u,v}\quad\mid\quad u\in U,\;\;v\in V\right\}.
  \]
  \begin{enumerate}
  \item Show that $N_q\subseteq N_{\tilde{\beta}}$, where $q\colon
    F_{U\times V}\to U\otimes V$ is the canonical quotient, and apply
    the universal property of quotient maps to get a map
    $\beta\colon U\otimes V\to W$.
  \item Show that $\alpha$ and $\beta$ are inverses by verifying that
    $\alpha\circ\beta=\Id_{U\otimes V}$ and $\beta\circ\alpha=\Id_W$.
  \end{enumerate}
  
  
  
\end{enumerate}

\end{document}

