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\pagestyle{fancy} \lhead{{\sf Homework 1 $|$ Due January 16 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear Algebra $|$
    Spring 2026 $|$ M.~Macauley}}


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\begin{enumerate}

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

\item Prove the following properties of a group $G$.
  \begin{enumerate}[(a)]
  \item The identity element $e\in G$ is unique. 
  \item The inverse of each $g\in G$ is unique. \medskip
  \end{enumerate}

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

\item Let $X$ be a vector space over a field $K$. Let $0$ be the zero
  element of $K$ and $\0$ the zero-element of $X$. Prove
  each of the following.
  \begin{enumerate}[(a)]
  \item $0x=\0$ for every $x\in X$;
  \item $k\0=\0$ for every $k\in K$; 
  \item For every $k\in K$ and $x\in X$, if $kx=\0$, then $k=0$ or
    $x=\0$. \medskip
  \end{enumerate}

%%------------------------------------------------------------------------
  
\item The \emph{characteristic} of a field $K$, denoted $\Char K$, is
  the smallest positive integer $n$ for which
  $n1:=\underbrace{1+1+\cdots+1}_{\tiny\mbox{$n$ times}}=0$, or zero
  if no such $n$ exists.
  \begin{enumerate}
  \item Show that if $K$ and $L$ are fields with $K\leq L$, then $L$
    is a vector space over $K$.
  \item Show that the characteristic of a field is either zero or prime. 
  \item Show that if $K$ is a finite field, then $|K|$ is a prime power.
  \item Show that if $K$ and $L$ are finite fields with $K\subset
    L$ and $|K|=p^m$ and $|L|=p^n$, then $m$ divides $n$. \medskip
  \end{enumerate}

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

\item Let $X$ be a vector space, and let $S$ be a linearly
  independent subset of $X$. The following is called the
  \emph{exchange property}: For any nonzero $x_0\in\Span(S)$, there
  exists $x_1\in S$ such that $S'=(S\setminus\{x_1\})\cup\{x_0\}$ is a
  basis for $\Span(S)$.
  \begin{enumerate}[(a)]
  \item Prove the exchange property.
  \item Suppose that $B$ is a size-$n$ basis for $X$, and let $B'$ be
    another basis. Use the exchange property to show that
    $|B'|=n$. \medskip
  \end{enumerate}
  
%%------------------------------------------------------------------------

\item Let $X_1$, $X_2$ be subspaces of a finite-dimensional vector
  space $X$. Show that $\dim(X_1\times X_2)=\dim(X_1\oplus
  X_2)$. \medskip

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

\item Let $Y$ be a subspace of $X$. Show that addition and scalar multiplication in the quotient $X/Y$ are \emph{well-defined}. That is, they do not depend on the choice of congruence class representatives. \medskip

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

\item Let $Y$ be a subspace of $X$. Prove that for any $x\in X$, the following two sets are equal as subsets of $X$:
      \[
      \{x\}=\{x+y\mid y\in Y\}.
      \]
      This motivates the alternative ``coset notation'' of $x+Y$ for the
      equivalence class of $x$ modulo $Y$. Show how to add and scalar
      multiply in this notation by computing:
      \[
      (x+Y)+(z+Y),\qquad\text{and}\qquad a(x+Y),\quad x,z\in X,\;a\in K.
      \]


\end{enumerate}

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