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\pagestyle{fancy} \lhead{{\sf Homework 1 $|$ Due August 24 (Wednesday)
}} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear Algebra $|$
    Fall 2016 $|$ M.~Macauley}}


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\noindent\emph{Read}: Lax, Chapter 1, pages 1--11. \\

\begin{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 $X$. Using only the
  definitions of a group, a vector space, and a field, carefully prove
  each of the following:
  \begin{enumerate}
  \item The identity element $e$ of a group is unique. 
  \item In any group $G$, the inverse of $g\in G$ is unique. 
  \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
    $x=\0$. \\
  \end{enumerate}
  
  
  
\item Let $X$ denote the vector space of polynomials in $\R[x]$ of
  degree less than $n$. Are the vectors $x^3+2x+5$, $3x^2+2$, $6x$,
  $6$ linearly independent in $X$? (Assume that $n\geq 4$.) \\
  
  
  
\item The following is called the \emph{Replacement Lemma}: Let $X$ be
  a vector space over $K$, and let $S$ be a linearly independent
  subset of $X$. Let $x_0\in\Span(S)$ with $x_0\neq 0$. Then there
  exists $x_1\in S$ such that the set
  $S'=(S\setminus\{x_1\})\cup\{x_0\}$ is a basis for $\Span(S)$.
  \begin{enumerate}
  \item Prove the Replacement Lemma.
  \item Suppose that $B$ is a basis for $X$ containing $n$ elements,
    and let $B'$ be another basis for $X$. Use the Replacement Lemma
    to show that $|B'|=n$. \\
  \end{enumerate}
  


\item Let $S$ be a set of vectors in a finite-dimensional vector space
  $X$. Show that $S$ is a basis of $X$ if every vector of $X$ can be
  written in one and only one way as a linear combination of the
  vectors in $S$. \\



\item Let $X_1$, $X_2$ be vector spaces over a field $K$. Show that
  $\dim(X_1\times X_2)=\dim X_1+\dim X_2$. \\

  

\item If $Y$ is a subspace of $X$, then two vectors $x_1,x_2\in X$ are
  \emph{congruent modulo $Y$}, denoted $x_1\equiv x_2\bmod Y$, if
  $x_1-x_2\in Y$. This is an equivalence relation; denote the
  equivalence class containing $x\in X$ by $\{x\}$, and let $X/Y$
  denote the set of equivalence classes. We can make $X/Y$ into a
  vector space by defining addition and scalar multiplication as
  follows:
  \[
  \{x\}+\{z\}=\{x+z\},\qquad \{ax\}=a\{x\}.
  \]
  \begin{enumerate}[(a)]
  \item Show that these operations are well-defined. That is, they
      do not depend on the choice of congruence class representatives.
  \item Assume $\dim X<\infty$. Show that $X$ is isomorphic to
    $Y\times X/Y$ by defining an explicit map and showing that it is
    linear and a bijection.
  \end{enumerate}

   
  


\end{enumerate}

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