
\documentclass{amsart}

\title{Math studies algebra Spring 2018: Homework 3}

\usepackage{amssymb} 

\usepackage{tikz-cd}

\author{JC}

\begin{document}

\maketitle

   This homework is due by class time on Monday 12 February. It must be typeset
   (preferably in\ \LaTeX) and submitted as a PDF file on the Canvas site,
   with a filename of the form \begin{verbatim} andrewID_alg_homeworknumber.pdf \end{verbatim}   
   For each minute that it is late, the grade will be reduced by 10 percent. 

\medskip

\begin{enumerate}

\item  Let $V$ be a VS over a field $k$. Prove that:

  \begin{enumerate}

  \item  A subset $X$ of $V$ is a basis of $V$ if and only if it is a {\em minimal spanning set}, that
    is to say $X$ is spanning but every proper subset $Y \subsetneq X$ fails to be spanning. 
  
  \item  If $X$ is spanning and $Y$ is a maximal element in the poset of independent subsets of $X$ (ordered by inclusion) then
      $Y$ is a basis.
  
  \item  If $X$ is independent and $Y$ is spanning then $\vert X \vert \le \vert Y \vert$. 

  \end{enumerate}
  
\item  Prove that if $W$ is a VS over $k$ and $V$ is a subspace of $W$ then for any basis $B$ of $V$ there is a basis
  $C$ of $W$ with $B \subseteq C$. Prove further that in this case the set $\{ c + V : c \in C \setminus B \}$ is
  a basis for the quotient space $W/V$. Use this to deduce that $\dim(W) = \dim(V) + \dim(W/V)$. 


\item  Let $0 \rightarrow V_0 \rightarrow \ldots \rightarrow V_n \rightarrow 0$ be an exact sequence
  of VS's over a field $k$ (so the maps from $V_i$ to $V_{i+1}$ are linear for $i < n$).
  Assume that $\dim(V_i)$ is finite for each $i$. Prove that
     $\sum_{i=0}^n (-1)^i \dim(V_i) = 0$. 

  
\item  Let $f: {\mathbb R} \rightarrow {\mathbb R}$ be such that $f(x + y) = f(x) + f(y)$ and $f(1) = 1$.

  \begin{enumerate}


  \item Prove that $f(q) = q$ for all rational $q$. 
    
  \item Prove that if $f$ is continuous then $f$ is the identity function.

  \item Show that there exists an $f$ as above which is not the identity function. Hint: ${\mathbb R}$ can be viewed
     as a VS over $\mathbb Q$. 

  \end{enumerate}  
  



\item  Let $P$ be a nonempty set of prime numbers. Let $S$ be a set of  integers such that
     for each $k \in S$, $k > 1$ and  only primes in $P$ appear in the prime factorisation of $k$. 
     Prove that if $\vert S \vert > \vert P \vert$ then there is a nonempty subset $S'$ of
     $S$ such that the product of the elements of $S'$ is a perfect square. Hint: Make it into a
     problem about linear independence in the  k-VS $k^{\vert P \vert}$ where $k = {\mathbb Z}/2 {\mathbb Z}$. 


\end{enumerate}  

     
\end{document} 
