\documentclass[12pt]{article}
\usepackage{latexsym}
\usepackage{amssymb}
\title{Algebra 1; MA20008; Sheet 3}
\author{{\tt G.C.Smith@bath.ac.uk}}
\date{25-x-2004}
\begin{document}
\maketitle
\begin{enumerate}
\item Suppose that $V$ is a vector space over $F$, and
that $S \subseteq V$. Let $\overline S$ be the intersection
of those subspaces of $V$ which contain the subset $S$, 
or put formally
\[ \overline S = \bigcap \{ U \mid U \leq V, S \subseteq U\}.\]
Show that $\overline S = \langle S  \rangle.$
\item  Let $V$ be a vector space over $F$
and ${\bf v_1}, {\bf v_2},\ldots, {\bf v_n} \in V$. Suppose that
whenever $\theta_1, \ldots, \theta_n, \psi_1, \ldots, \psi_n \in F$
and $\sum_{i=1}^n \theta_i {\bf v_i} = \sum_{i=i}^n \psi_i {\bf v_i}$,
then necessarily $\theta_i = \psi_i$ for each $i$, $1 \leq i \leq n$.
Show that ${\bf v_1}, {\bf v_2},\ldots, {\bf v_n} \in V$ is
linearly independent.
\item Consider $V = \mathbb C$ as a vector space over $\mathbb Q$.
\begin{enumerate}
\item Show that $1, \sqrt 2, \sqrt 3$ are linearly independent.
\item Let $\alpha = e^{\frac{\pi i}{3}}$. Which lists
of the form $1, \alpha, \ldots, \alpha^n$ are linearly independent?
Justify your answer.
\item Suppose that $1, \beta, \beta^2, \ldots, \beta^n$ are 
linearly independent. Show that
$1, (\beta +1), (\beta +1)^2, \ldots,  (\beta +1)^n$ are linearly independent. 
\end{enumerate}
\item Suppose that $X$ and $Y$ are both linearly independent subsets
of $V$. Does it follow that $X \cap Y$ is linearly independent?
What about $X \cup Y$?
\item Suppose that $V = U \oplus W$. We are given
a sets of vectors $X \subseteq U$
and $Y  \subseteq W$. Is $X \cup Y$ necessarily a linearly
independent set of vectors?
\item Suppose that ${\bf v_1}, {\bf v_2},\ldots, {\bf v_n}$ is a linearly
independent list of vectors in the vector space $V$. We are given 
${\bf w} \in V$. Does it follow that
\[{\bf v_1}+ {\bf w},  {\bf v_2}+ {\bf w}, \ldots, {\bf v_n} + {\bf w}\]
are linearly independent?
\item Suppose that ${\bf v_1}, {\bf v_2},\ldots, {\bf v_n}$ is a linearly
dependent list of vectors in the vector space $V$. We are given
${\bf w} \in V$. Does it follow that
\[{\bf v_1}+ {\bf w},  {\bf v_2}+ {\bf w}, \ldots, {\bf v_n} + {\bf w}\]
is linearly dependent?
\item Let $V = \mathbb R^3$ viewed as vector space over $\mathbb R$.
Let ${\bf v_1}, \ldots, {\bf v_8}$ be the position vectors of the vertices
of a cube.
\begin{enumerate}
\item Let 
\[ A = \left\{ \sum_i \lambda_i {\bf v_i} \mid 0 \leq \lambda_i \leq 1 \mbox{ for all }
i, \sum_i \lambda_i = 1\right\}. \]
Describe the set $A$, viewed as a collection of position vectors, geometrically.
\item  Let
\[ B = \left\{ \sum_i \lambda_i {\bf v_i}  \mid \lambda_i \geq 0 \mbox{ for all } i, 
\right\}. \]
Under what circumstances is $B = \mathbb R^3$? Under what circumstances
is $B$ a closed half space (i.e. one side of a plane and all the
points on that plane)? What other shapes can arise?
\end{enumerate}
\end{enumerate}
\end{document}