\chapter{Subspaces of $(\mathbb{R}^m,+,\cdot)$}
\markboth{Lecture \thechapter : Subspaces of $(\mathbb{R}^m,+,\cdot)$}
         {Lecture \thechapter : Subspaces of $(\mathbb{R}^m,+,\cdot)$}
\label{lecture1}

\section{Closed set \important}
\label{sec1.1}

A set ${\cal S}$ of vectors of $(\mathbb{R}^m,+,\cdot)$ is said to be
\textit{closed under +} if and only if the sum of any two vectors of ${\cal S}$ yields a vector of $\mathcal{S}$. The set ${\cal S}$ is {\it closed under $\cdot$} if and only if any real multiple of a vector in ${\cal S}$ also belongs to ${\cal S}$. Mathematically:
\[ \mathcal{S} \text{ closed under } + \quad \Leftrightarrow \quad
   \forall\a\in\S,\forall\b\in\S,\;\a+\b\in\S \]
and
\[ \S \text{ closed under } \cdot \quad \Leftrightarrow \quad
   \forall\a\in\S, \alpha\cdot\a\in\S. \]
Consider $n$ vectors $\a_1,\ldots,\a_n$ in $\mathbb{R}^m$ and $n$ real numbers $\alpha_1,\ldots,\alpha_n$. The vector
\begin{equation} \label{eq1}
 \b = \alpha_1\cdot\a_1+\ldots+\alpha_n\cdot\a_n 
\end{equation}
is called a {\it linear combination} of the vectors $\a_1,\ldots,\a_n$.
\marginpar{\shadowbox{Linear combination}}

The linear combination (\ref{eq1}) is well-defined in the sense that 
it is independent of the order in which it is evaluated, thanks to the 
vector space property (P2) of $(\mathbb{R}^m,+,\cdot)$.
The vector 
\begin{equation} \label{eq2}
\begin{bmatrix}
  \alpha_1\\
  \vdots\\
  \alpha_n
\end{bmatrix}
\equiv \b_{\S}
\end{equation}
of coefficients of the vector $\b$ as a linear combination of the
\marginpar{\shadowbox{Vector representation}}
vectors $\a_1,\ldots,\a_n$ is called a
\textit{representation} of $\b$ with respect to the set 
$\S=\{\a_1,\ldots,\a_n\}$.

\begin{example}
\textrm{\small
We have seen in Lecture ... 
}
\end{example}

\begin{figure}[t]
\begin{picture}(320,225)(-10,45)

\put(30,225){\makebox(0,0)[t]{
\fbox{$2\cdot\a_1=\begin{bmatrix}2\\-2\end{bmatrix}$}}}
\put(110,225){\makebox(0,0)[t]{
\fbox{$(-1)\cdot\a_2=\begin{bmatrix}0\\-2\end{bmatrix}$}}}
\put(30,194){\line(0,-1){19}}
\put(110,194){\line(0,-1){19}}

\put(70,175){\makebox(0,0)[t]{
\fbox{$2\cdot\a_1+(-1)\cdot\a_2=\begin{bmatrix}2\\-4\end{bmatrix}$}}}
\put(190,175){\makebox(0,0)[t]{
\fbox{$3\cdot\a_3=\begin{bmatrix}-3\\6\end{bmatrix}$}}}
\put(70,144){\line(0,-1){19}}
\put(190,144){\line(0,-1){19}}

\put(130,125){\makebox(0,0)[t]{
\fbox{$(2\cdot\a_1+(-1)\cdot\a_2)+3\cdot\a_3=\begin{bmatrix}-1\\2\end{bmatrix}$}}}
\put(270,125){\makebox(0,0)[t]{
\fbox{$2\cdot\a_4=\begin{bmatrix}2\\2\end{bmatrix}$}}}
\put(130,94){\line(0,-1){19}}
\put(270,94){\line(0,-1){19}}

\put(200,75){\makebox(0,0)[t]{\fbox{
        $\b=(((2\cdot\a_1+(-1)\cdot\a_2)+3\cdot\a_3)+2\cdot\a_4
            =\begin{bmatrix}1\\4\end{bmatrix}$}}}
\end{picture}
\caption{Sequential summation of vectors.}
\label{fig1}
\end{figure}

\section*{Problems}

\begin{problems}
\item \label{pb1}
Show that the vector 
\begin{equation} \label{galrep}
 \b_\S = \begin{bmatrix}
          t-s+1\\
          -\frac{t}{2}-s+\frac{5}{2}\\
          t\\
          s
         \end{bmatrix} 
\end{equation}
is a representation ...
\item How many ``levels'' of operations would the evaluation of a linear combination of $n=8$ vectors require if 
\begin{itemize}
\item[a.] a sequential process is used?
\item[b.] a 2-processor parallel process is used?
\end{itemize}
What is the ratio sequential/parallel between the two numbers (speed-up)? 
\end{problems}