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\title{MA20008 Algebra 1, 2004, Sheet 9}
\author{Geoff Smith, {\tt http://www.bath.ac.uk/$\sim$masgcs}}
\date{}
\begin{document}
\maketitle
\begin{enumerate}
\item Let $U$ be the set of polynomials in the variable
$X$ with coefficients in $\mathbb R$. We define an inner product
$\langle \ ,\ \rangle$ on $U$ via
\[ \langle f, h \rangle = \int_0^1 fh\ dX.\]
Thus $U$ is a vector space of $\mathbb R$ in the natural way.
Let $V$ be the subspace of $U$ consisting of polynomials of degree at
most 3. 
Given the basis $1, X, X^2, X^3$, run the Gram-Schmidt algorithm
to produce an orthonormal basis of $V$. 
\item Let $V$ be an inner product space with orthonormal basis
${\bf v_1}, {\bf v_2}, \ldots, {\bf v_n}$. If the Gram-Schmidt algorithm
is used to modify this basis, what is the output?
\item Let $V$ be an inner product space with orthonormal basis
${\bf v_1}, {\bf v_2}, \ldots, {\bf v_n}$. We obtain another
basis ${\bf v_n}, {\bf v_{n-1}}, \ldots, {\bf v_1}$ by reversing
the order of the vectors. Run the Gram-Schmidt algorithm on each
of these bases in turn. Is it true that the output orthonormal bases
are the reverse of each other?
\item For $r= 0,1,2$ define functions $f_r : \mathbb R \longrightarrow \mathbb R$
by $f_r : \theta \mapsto \cos r\theta$ for all real numbers $\theta$.
The collection of all functions $f: \mathbb R \longrightarrow \mathbb R$
has a natural vector space structure. Let $V$ be
the subspace spanned by $f_0, f_1, f_2$. Define an inner product on
$V$ via $\langle f, h \rangle = \frac{1}{2\pi}\int_0^{2\pi} fh\ d\theta.$
Run Gram-Schmidt to obtain an orthonormal basis of $V$.
\item Suppose that ${\bf e_1}, {\bf e_2}, \ldots,  {\bf e_n}$ is an orthonormal
basis of the inner product space $V$. Let $U_r = \langle {\bf e_1}, {\bf e_2}, \ldots,  
{\bf e_r} \rangle.$
Suppose that  ${\bf v} \in V$. Show that among all vectors
${\bf x} \in U_r$, the one which minimizes $||{\bf v} - {\bf x}||$
is $\sum_{i=1}^r \langle {\bf v}, {\bf e_i} \rangle {\bf e_i}.$
\item Suppose that $V$ is an inner product space of dimension $n$,
and $\alpha : V \longrightarrow V$ is a linear map. Suppose that
$\alpha$ carries some orthonormal basis to an orthonormal basis.
Show that $\alpha$ carries each orthonormal basis to an orthonormal
basis.
\end{enumerate} 
\end{document}
