\documentclass[12pt]{article}
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\title{Algebra 1: Math20008: Sheet 1}
\author{\tt G.C.Smith@bath.ac.uk}
\date{October 12, 2004}
\begin{document}
\maketitle

{\it The course web site is available via 
{\tt http://www.bath.ac.uk/$\sim$masgcs/}}

\begin{enumerate}

\item Let $V$ be a vector space over $F$. {\em Here is a proof that
$0 {\bf v} = {\bf 0}$ for all ${\bf v} \in V$. First note that
$0+0 = 0$ (Field Axiom 3). Therefore 
$(0+0){\bf v} = 0{\bf v}$. Now by Vector Space Axiom 7 we have
$0{\bf v} + 0{\bf v} = 0{\bf v}.$ Now, by Vector Space Axiom 4
the vector $0{\bf v}$ has an additive inverse $-0{\bf v}$.
It follows that
\[ (0{\bf v} + 0{\bf v}) + -0{\bf v} = 0{\bf v} + -0{\bf v}.\]
Now rebracket the left hand side using Vector Space Axiom 2.
Therefore
\[ 0{\bf v} + (0{\bf v} + -0{\bf v}) = 0{\bf v} + -0{\bf v}.\]
Now on both sides we have the opportunity to use the definition
of an additive inverse so
$0{\bf v} + {\bf 0} = {\bf 0}.$
Now by Vector Space Axiom 3 we deduce that
\[ 0{\bf v} = {\bf 0}.\]} Now, using this as a model, and 
quoting this result if necessary, prove that if ${\bf v} \in V$,
then $(-1){\bf v} = -{\bf v}.$
\item Let $V$ be a vector space over $F$. Suppose that $U$ is a subset 
of $V$. Show that $U$ is a subspace of $V$ if and only
if the following three conditions are satisfied:
\begin{enumerate}
\item $U \not = \emptyset$.
\item If ${\bf x} \in U$ and $\lambda \in F$, then $\lambda{\bf x} \in U$.
\item If ${\bf x},\ {\bf y} \in U,$ then ${\bf x}+{\bf y} \in U.$
\end{enumerate}
\item Consider the usual Cartesian description
of the plane as ${\mathbb R}^2$ (with perpendicular axes).
This collection of ordered pairs is a vector space
over $\mathbb R$ in a natural way as discussed
in lectures. In each case you should justify your answer.
\begin{enumerate}
\item[(a)] Prove that the ordered pairs corresponding a straight
line through the origin form a subspace.
\item[(b)] Consider the straight line S = $\{(x,1) \mid x \in \mathbb R \}.$
Is this a subspace?
\item[(c)] Consider the circle $C = \{ (x, y) \mid x^2 + (y-1)^2 = 1 \}.$
Is this a subspace?
\item[(d)] Prove that $Z = \{ (0,0)\} $ is a subspace.
\item Prove that $\emptyset$ is not a subspace.
\end{enumerate}
\item Describe as many different subspaces of $\mathbb R^3$ as you can find.
\item Let $\mathbb R[X]$ denote the set of polynomials in $X$ which
have coefficients in $\mathbb R$. This set has a natural vector
space structure over $\mathbb R$. Which of the following are subspaces
of $\mathbb R[X]$, and why?
\begin{enumerate}
\item[(a)] $\{ f \mid f \in \mathbb R[X], f(42) = 0\}$.
\item[(b)] $\{ f \mid f \in \mathbb R[X], f(42) = 1\}$.              
\item[(c)] $\{ f \mid f \in \mathbb R[X], f \mbox{ has at most two real roots }\}$.              
\item[(d)] $\{ f \mid f \in \mathbb R[X], \deg f \leq n\}$ where $n \in \mathbb N \cup
\{0\}.$ Note that the degree of the zero polynomial is $-\infty$, a symbol
deemed to be smaller than all integers. 
\item[(e)] $\{ f \mid f \in \mathbb R[X], f''(X) - Xf'(X) + f(X) = 0\}$
where a dash in the exponent indicates differentiation with respect to $X$. 
\item[(f)] $\{ f \mid f(X)^2 = f(X^2)\}.$             
\end{enumerate}
\item Suppose that $V$ is a vector space and that ${\bf v}_1, {\bf v}_2, \ldots,
{\bf v}_m \in V.$ Let \[U = 
\left\{ \sum_{i=1}^n \lambda_i {\bf v}_i \Biggm\vert \lambda_i \in \mathbb R
\mbox{ for all } 1 \leq i \leq n\right\}.\] Show that $U$ is a subspace of $V$.
\item Suppose that $U, W$ are subspaces of $V$ and that
$U \cup W$ is also a subspace of $V$.
Prove that either $U \subseteq W$ or $W \subseteq U$.
\end{enumerate} 
\end{document}
 
