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%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{Introduction to Vector Spaces\label{Introduction to Vector Spaces}}

\subsubsection{Vector Space\label{vector space}}

A collection $V$ of $\hyperref{n-vectors}{}{}{MA01_04.tex#n-vector}$ is a 
\textbf{vector space} if $V$ satisfies the following properties:

\begin{enumerate}
\item  (\textit{Closure under vector addition}) The sum of two vectors $u$
and $v$ in $V$ is a vector $u+v$ that belongs to $V$.

\item  (\textit{Closure under scalar multiplication}) The product of a
vector $v$ in $V$ with any scalar $c$ is a vector $cv$ that belongs to $V$.
\end{enumerate}

\noindent If $V$ satisfies these properties, it is said to be \textbf{closed}
under vector addition and scalar multiplication.

\subsubsection{Subspace\label{subspace}}

A vector space $V$ is a \textbf{subspace} of a vector space $W$ if every
vector in $V$ also belongs to $W$.

\begin{definition}
Let $A$ be an $m\times n$ matrix.

\begin{enumerate}
\item  The subspace of $\UNICODE{0x211d} ^{n}$ consisting of all solutions
to the \hyperref{homogeneous linear system}{}{}{MA01_03.tex#homogeneous} $%
Ax=0$ is called the \textbf{null space\label{null space}} of the matrix $A$,
and is denoted by $N\left( A\right) $.

\item  The subspace of $\UNICODE{0x211d} ^{m}$ consisting of all vectors $b$
for which the system $Ax=b$ is \hyperref{consistent}{}{}{%
MA01_01.tex#consistent} is called the \textbf{column space\label%
{column space}} of the matrix $A$, and is denoted by $C\left( A\right) $.
\end{enumerate}
\end{definition}

\begin{example}
Let $A=\left[ 
\begin{array}{ccc}
1 & 2 & 3 \\ 
3 & 6 & 9
\end{array}
\right] \ $. The null space of $A$ is the subspace of  $\UNICODE{0x211d} ^{3}
$ consisting of all vectors of the form  
\[
\left[ 
\begin{array}{ccc}
2s+3t & -s & -t
\end{array}
\right] 
\]
for real numbers $s$ and $t$. The column space of $A$  is the subspace of $%
\UNICODE{0x211d} ^{2}$ consisting of all vectors of the form 
\[
\left[ 
\begin{array}{cc}
a & 3a
\end{array}
\right] 
\]
for real numbers $a$. (Observe that this is the same as saying all vectors
of the form 
\[
s\left[ 
\begin{array}{ccc}
2 & -1 & 0
\end{array}
\right] +t\left[ 
\begin{array}{ccc}
3 & 0 & -1
\end{array}
\right] 
\]
and 
\[
a\left[ 
\begin{array}{cc}
1 & 3
\end{array}
\right] 
\]
respectively.)
\end{example}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Column space}{}{}{../../Help/DM8-7.tex#Column space}}

{\small \hyperref{Nullspace}{}{}{../../Help/DM8-7.tex#Nullspaces}}

\noindent \hrulefill

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