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<main class="main"><div id="content" class="pretext-content"><section xmlns:svg="http://www.w3.org/2000/svg" class="section" id="section-25"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">4.1</span> <span class="title">Initial definitions</span>
</h2>
<section class="subsection" id="subsection-56"><h3 class="heading hide-type">
<span class="type">Subsection</span> <span class="codenumber">4.1.1</span> <span class="title">The definitions of Euclidean \(n\)-space</span>
</h3>
<p id="p-722">Euclidean \(n\) space, also called \(\R^n\text{,}\) is formally written as</p>
<div class="displaymath">
\begin{equation*}
\{(x_1,x_2,\ldots,x_n)\mid x\in\R\}.
\end{equation*}
</div>
<p class="continuation">This means that it consists of elements called <dfn class="terminology">\(n\)-tuples</dfn>, which are written as \((x_1,x_2,\ldots,x_n)\) where each \(x_i\) is a real number. Each such element is called a <dfn class="terminology">vector</dfn>.</p>
<article class="example example-like" id="example-39"><a data-knowl="" class="id-ref example-knowl original" data-refid="hk-example-39"><h6 class="heading">
<span class="type">Example</span><span class="space"> </span><span class="codenumber">4.1.1</span><span class="period">.</span><span class="space"> </span><span class="title">Examples of \(n\)-tuples in \(\R^n\).</span>
</h6></a></article><div class="hidden-content tex2jax_ignore" id="hk-example-39"><article class="example example-like"><ul id="p-723" class="disc">
<li id="li-260"><p id="p-724">\((2,3)\) is in \(\R^2\)</p></li>
<li id="li-261"><p id="p-725">\((-3,0,\frac12)\) is in \(\R^3\)</p></li>
<li id="li-262"><p id="p-726">\((-1,0,4,\sqrt2,\frac\pi2,1000)\) is in \(\R^6\)</p></li>
</ul></article></div>
<p id="p-727">The most familiar examples, of course, are \(\R^2\text{,}\) the plane, and \(\R^3\text{,}\) ordinary 3-dimensional space. For \(\R^2\) we have an \(x\)-axis and a \(y\)-axis, and the points in the plane as \(2\)-tuples are defined by dropping perpendiculars to each axis.</p>
<figure class="figure figure-like" id="figure-14"><div class="image-box" style="width: 50%; margin-left: 25%; margin-right: 25%;"><div class="asymptote-box" style="padding-top: 99.6966238652069%"><iframe src="images/image-14.html" class="asymptote"></iframe></div></div>
<figcaption><span class="type">Figure</span><span class="space"> </span><span class="codenumber">4.1.2<span class="period">.</span></span><span class="space"> </span></figcaption></figure><p id="p-728">Euclidean \(3\)-space is viewed analogously.</p>
<figure class="figure figure-like" id="figure-15"><div class="image-box" style="width: 60%; margin-left: 20%; margin-right: 20%;"><div class="asymptote-box" style="padding-top: 100.497512437811%"><iframe src="images/image-15.html" class="asymptote"></iframe></div></div>
<figcaption><span class="type">Figure</span><span class="space"> </span><span class="codenumber">4.1.3<span class="period">.</span></span><span class="space"> </span></figcaption></figure><p id="p-729">There are, of course, many geometric concepts studied in \(\R^2\) and \(\R^3\text{.}\) One of our goals is to see how these concepts can be extended to \(\R^n\text{.}\)</p>
<p id="p-730">While Euclidean \(n\)-space consists of \(n\)-tuples, they are sometimes viewed from different mathematical perspectives.</p>
<ul class="disc">
<li id="li-263"><p id="p-731">Points in \(n\)-space: the vectors are just the \(n\)-tuples \((x_1,x_2,\ldots,x_n)\text{.}\)</p></li>
<li id="li-264"><p id="p-732">Directed vectors: the vectors may be thought of as arrows from an initial point \(P\) to a terminal point \(Q\text{.}\)</p></li>
</ul>
<figure class="figure figure-like" id="figure-16"><div class="image-box" style="width: 60%; margin-left: 20%; margin-right: 20%;"><div class="asymptote-box" style="padding-top: 79.0050935048518%"><iframe src="images/image-16.html" class="asymptote"></iframe></div></div>
<figcaption><span class="type">Figure</span><span class="space"> </span><span class="codenumber">4.1.4<span class="period">.</span></span><span class="space"> </span></figcaption></figure><ul id="p-733" class="disc">
<li id="li-265">
<p id="p-734">Column vectors where the vector is an \(n\times 1\) matrix:</p>
<div class="displaymath">
\begin{equation*}
\begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_n \end{bmatrix}
\end{equation*}
</div>
</li>
<li id="li-266">
<p id="p-735">Row vectors where the vector is a \(1\times n\) matrix:</p>
<div class="displaymath">
\begin{equation*}
\begin{bmatrix} x_1, x_2, \ldots, x_n \end{bmatrix}
\end{equation*}
</div>
</li>
</ul>
<p class="continuation">For our initial discussion, we will concentrate on \(n\)-tuples and column vectors.</p></section><section class="subsection" id="subsection-57"><h3 class="heading hide-type">
<span class="type">Subsection</span> <span class="codenumber">4.1.2</span> <span class="title">Equality, addition and scaler multiplication of vectors</span>
</h3>
<p id="p-736">The equality, addition and scalar multiplication of \(n\)-tuples is very much like that of matrices:</p>
<ul class="disc">
<li id="li-267"><p id="p-737">Equality: \((x_1,x_2,\ldots,x_n)=(y_1,y_2,\ldots,y_n)\) means \(x_i=y_i\) for \(i=1,2,\ldots,n\text{.}\)</p></li>
<li id="li-268"><p id="p-738">Addition: \((x_1,x_2,\ldots,x_n)+(y_1,y_2,\ldots,y_n)
=(x_1+y_1,x_2+y_2,\ldots,x_n+y_n)\text{.}\)</p></li>
<li id="li-269"><p id="p-739">Scalar multiplication: For any scalar \(r\text{,}\) \(r(x_1,x_2,\ldots,x_n) =(rx_1,rx_2,\ldots,rx_n)\text{.}\)</p></li>
</ul>
<p class="continuation">In fact, if we look at the \(n\)-tuples as column vectors, then equality, addition and scalar multiplication are the same as matrix equality, addition and scalar multiplication.</p>
<article class="theorem theorem-like" id="theorem-46"><h6 class="heading">
<span class="type">Theorem</span><span class="space"> </span><span class="codenumber">4.1.5</span><span class="period">.</span><span class="space"> </span><span class="title">First properties of \(n\)-tuples.</span>
</h6>
<p id="p-740">Let \(\vec x=(x_1,\ldots,x_n)\text{,}\) \(\vec y=(y_1,\ldots,y_n)\text{,}\) and \(\vec z=(z_1,\ldots,z_n)\) be vectors in \(\R^n\text{,}\) and let \(r\) and \(s\) be scalars. In addition, let \(\vec 0=(0,\ldots,0)\) and \(-\vec x=(-x_1,-x_2,\ldots,-x_n)\text{.}\) Then</p>
<figure class="table table-like" id="table-7"><figcaption><span class="type">Table</span><span class="space"> </span><span class="codenumber">4.1.6<span class="period">.</span></span><span class="space"> </span></figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr>
<td class="l m b0 r0 l0 t0 lines">(A\(_1\)) \(\phantom{|}\vec x + \vec y\) is in \(\R^n\)</td>
<td class="l m b0 r0 l0 t0 lines">(M\(_1\)) \(\phantom{|}r\vec x\) is in \(\R^n\)</td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">(A\(_2\)) \(\phantom{|}\vec x + (\vec y + \vec x)
=(\vec x + \vec y) + \vec x\phantom{xx}\)</td>
<td class="l m b0 r0 l0 t0 lines">(M\(_2\)) \(\phantom{|}r(\vec x+\vec y)=r\vec x+r\vec y\)</td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">(A\(_3\)) \(\phantom{|}\vec x + \vec 0 = \vec x\)</td>
<td class="l m b0 r0 l0 t0 lines">(M\(_3\)) \(\phantom{|}(r+s)\vec x =r\vec x+s\vec x\)</td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">(A\(_4\)) \(\phantom{|}\vec x+ (-\vec x) = \vec 0\)</td>
<td class="l m b0 r0 l0 t0 lines">(M\(_4\)) \(\phantom{|}(rs)\vec x =r(s\vec x)\)</td>
</tr>
<tr>
<td class="l m b0 r0 l0 t0 lines">(A\(_5\)) \(\phantom{|}\vec x + \vec y = \vec y + \vec x \)</td>
<td class="l m b0 r0 l0 t0 lines">(M\(_5\)) \(\phantom{|}1\vec x =\vec x\)</td>
</tr>
</table></div></figure></article><article class="hiddenproof" id="proof-60"><a data-knowl="" class="id-ref proof-knowl original" data-refid="hk-proof-60"><h6 class="heading"><span class="type">Proof<span class="period">.</span></span></h6></a></article><div class="hidden-content tex2jax_ignore" id="hk-proof-60"><article class="hiddenproof"><p id="p-741">If we view each vector as a column vector, then each of the statements have been proven already in our study of matrix theory. (see <a class="xref" data-knowl="./knowl/MatrixAdditionProperties.html" title="Theorem 2.2.2: Addition properties of Matrices">Theorem 2.2.2</a> and <a class="xref" data-knowl="./knowl/ScalarMultiplicationProperties.html" title="Theorem 2.3.4: Properties of scalar multiplication">Theorem 2.3.4</a>). There is no need to do it again!</p></article></div></section></section></div></main>
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