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\title{Linear Vector Space}
\author{Kristen Zybura}
\maketitle
\tableofcontents
\section{Linear Vector Space Definition}
Let $V$ be a set on which two operations, addition and scalar multiplication, have been defined. If \textbf{$u$} and \textbf{$v$} are in $V$, the sum of \textbf{$u$} and \textbf{$v$} is denoted by \textbf{$u+v$}, and if $c$ is a scalar, the scalar multiple of \textbf{$u$} by $c$ is denoted by $c\textbf{u}$. If the following axioms hold for all $\textbf{u,v}$, and $\textbf{w}$ in $V$ and for all scalars $c$ and $d$, then $V$ is called a \textit{vector space} and its elements are called vectors.
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\textbf{10 Axioms of a Vector Space}

1. \textbf{u} $+$ \textbf{v} is in $V$ (closure under addition)

2. \textbf{u} $+$ \textbf{v} $=$ \textbf{v} $+$ \textbf{u} (commutativity)

3. (\textbf{u} $+$ \textbf{v}) $+$ \textbf{w} $=$ \textbf{u} $+$ (\textbf{v} $+$ \textbf{w}) (associativity)

4. There exists an element \textbf{0} in $V$, called a \textbf{zero vector}, such that \textbf{u}$+$\textbf{0}$=$\textbf{u}.

5. For each \textbf{u} in $V$, there is an element \textbf{$-$u} in $V$ such that \textbf{u} $+$ ($-$\textbf{u})$=$\textbf{0}.

6. $c$\textbf{u} is in $V$ (closure under scalar multiplication)

7. $c$(\textbf{u} $+$ \textbf{v}) $=$ $c$\textbf{u} $+$ $c$\textbf{v} (distributivity)

8. ($c + d$)\textbf{u} $= c$\textbf{u} $+ d$\textbf{u} (distributivity)

9. $c$($d$\textbf{u}) $=$ ($cd$)\textbf{u}

10. 1\textbf{u} $=$ \textbf{u}

\section{\textbf{Example}}
\textbf{Determine whether the given set is a vector space:}

\textbf{Let $V$ be the set of all positive real numbers $x$ with $x +x' = xx'$, and $kx=x^k$}

1. If $x$ and $y$ are positive reals, so is $x+y=xy$ ( closure under addition)

2. $x+y=xy=yx=y+x$ (commutativity)

3. $x +(y+z)=x(yz)=(xy)z=(x+y)+z$ (associativity)

4. $1+x=1x=x=x1=x+1$ for all positive reals $x$ (zero vector)

5. $x+(\frac{1}{x})=x(\frac{1}{x})=1=\textbf{0}=1=(\frac{1}{x})x=(\frac{1}{x})+x$     $c$\textbf{u} is in $V$ (closure under scalar multiplication)

6. If $k$ is real and $x$ is a positive real, then $kx=x^k$ is again a positive real.

7. $k(x+y)=(xy)^k=x^ky^k=kx+ky$ (distributivity)

8. $(c+d)x=x^{c+d}=x^cx^d=cx+dx$ (distributivity)

9. $c(dx)=(dx)^c=(x^d)^c=x^{dc}=x^{cd}=(cd)x$

10. $1x=x^1=x$


\section{Linear Combination}

A vector $v$ is a \textit{linear combination} of vectors $v_1,v_2,...,v_k$ if there are scalar coefficients $c_1,c_2,...,c_k$ such that $c_1v_1+c_2v_2+c_kv_k=v$

\section{Example of a linear combination}

\textbf{$u$}$=[1,2,-1]$   \textbf{$v$}$=[6,4,2]$ in $R^3$.

\textbf{Show that \textbf{w}$=[9,2,7]$ is a linear combination of \textbf{$u$} and \textbf{$v$}}.

$c_1[1,2,-1] + c_2[6,4,2]=[9,2,7]$

$[c_1,2c_1,-1c_1]+[6c_2,4c_2,2c_2]=[9,2,7]$

$c_1+6c_2=9$

$2c_1+4c_2=2$

$-c_1+2c_2=7$

$c_1=-3$

$c_2=2$

\section{Linear Dependence}

A set of vectors $v_1,v_2,...v_k$ is \textit{linearly dependent} if there are scalars $c_1,c_2,...,c_k$ at least one of which is not zero, such that $c_1v_1+c_2v_2+...c_kv_k=0$

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\textbf{ie} We say that a set of vectors is \textbf{linearly dependent} if one of them can be written as a linear combination of the others. 

A set of vectors which is not linearly dependent is \textbf{linearly independent}.

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\textbf{Example:}

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Let \textbf{u}$=[1,0,3]$ \textbf{v}$=[-1,1,-3]$ \textbf{w}$=[1,2,3]$

$3[1,0,3] + 2[-1,1,-3] - [1,2,3] = 0$

\textbf{u, v}, and \textbf{w} are linearly dependent, since

3\textbf{u} + 2\textbf{v} - \textbf{w} = 0  in fact, all the scalars are nonzero.

\medskip

\textbf{Definition:} If $S={v_1,v_2,...,v_k}$ is a set of vectors in a vector space $V$, then the set of all linear combinations of $v_1,v_2,...,v_k$ is called the \textit{span} of $v_1,v_2,...,v_k$.

\textbf{The Basis Theorem:} If a vector space V has a basis with \textit{n} vectors, then every basis for $V$ has exactly $n$ vectors.

\textbf{Definition:} A vector space $V$ is called \textit{finite dimensional} if it has a basis consisting of finitely many vectors. The \textit{dimension} of $V$, denoted by dim $V$, is the number of vectors in a basis for $V$. The dimension of the zero vector space {\textbf{0}} is defined to be zero. A vector space that has no finite basis is called \textit{infinite-dimensional}.

\textbf{Vector Addition:} \textbf{$u + v = [u_1 + v_1, u_2 + v_2,..., u_n + v_n]$}

\textbf{Scalar Multiplication:} $c\textbf{v} = c[v_1,v_2]=[cv_1,cv_2]$

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