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<p style='margin-left:36.0pt'><span lang=EN-US><!--[if gte vml 1]><v:shape
 id="_x0000_i1045" type="#_x0000_t75" alt="" style='width:103.5pt;height:41.25pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image020.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_263.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=138 height=55
src="./第六章%20附录——图形变换.files/image020.gif" v:shapes="_x0000_i1045"><![endif]></span></p>

<p style='margin-left:36.0pt;line-height:200%'>下面让我们用一个简单的例子来说明，设<i><span
lang=EN-US style='font-family:"Times New Roman"'>A</span></i>为<span lang=EN-US>2x3的矩阵，</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>B</span></i>为<span lang=EN-US>3x2的矩阵，则两者的乘积为：</span></p>

<p><sub><span lang=EN-US>&nbsp;&nbsp;&nbsp; <!--[if gte vml 1]><v:shape id="_x0000_i1046"
 type="#_x0000_t75" alt="" style='width:372.75pt;height:121.5pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image021.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_264.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=497 height=162
src="./第六章%20附录——图形变换.files/image021.gif" v:shapes="_x0000_i1046"><![endif]><o:p></o:p></span></sub></p>

<p><span lang=EN-US>&nbsp; 4) 单位矩阵 </span></p>

<p style='margin-left:36.0pt;line-height:200%'>对于一个<i><span lang=EN-US
style='font-family:"Times New Roman"'>n</span></i><span lang=EN-US>x</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>n</span></i>的矩阵，如果它的对角线上的各个元素均为<span
lang=EN-US>1，其余元素都为0，则该矩阵称为<span style='color:maroon'>单位矩阵</span>，记为</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>I<sub>n</sub></span></i>。对于任意<i><span
lang=EN-US style='font-family:"Times New Roman"'>m</span></i><span lang=EN-US>x</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>n</span></i>的矩阵恒有</p>

<p align=center style='margin-left:36.0pt;text-align:center'><span lang=EN-US><!--[if gte vml 1]><v:shape
 id="_x0000_i1047" type="#_x0000_t75" alt="" style='width:91.5pt;height:42.75pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image022.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_265.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=122 height=57
src="./第六章%20附录——图形变换.files/image022.gif" v:shapes="_x0000_i1047"><![endif]></span></p>

<p><span lang=EN-US>&nbsp; 5) 矩阵的转置 </span></p>

<p style='margin-left:36.0pt;line-height:200%'>交换一个矩阵<i><span lang=EN-US
style='font-family:"Times New Roman"'>A<sub>m</sub></span></i><sub><span
lang=EN-US>x</span></sub><i><sub><span lang=EN-US style='font-family:"Times New Roman"'>n</span></sub></i>的所有的行列元素，那么所得到的<i><span
lang=EN-US style='font-family:"Times New Roman"'>m</span></i><span lang=EN-US>x</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>n</span></i>的矩阵被称为原有矩阵的<span
style='color:maroon'>转置</span>，记为<i><span lang=EN-US style='font-family:"Times New Roman"'>A<sup>T</sup></span></i>：</p>

<p align=center style='margin-left:36.0pt;text-align:center'><span lang=EN-US><!--[if gte vml 1]><v:shape
 id="_x0000_i1048" type="#_x0000_t75" alt="" style='width:137.25pt;height:55.5pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image023.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_266.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=183 height=74
src="./第六章%20附录——图形变换.files/image023.gif" v:shapes="_x0000_i1048"><![endif]></span></p>

<p style='margin-left:36.0pt'><span lang=EN-US><!--[if gte vml 1]><v:shape
 id="_x0000_i1049" type="#_x0000_t75" alt="" style='width:258.75pt;height:19.5pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image024.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_267.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=345 height=26
src="./第六章%20附录——图形变换.files/image024.gif" v:shapes="_x0000_i1049"><![endif]>&nbsp;&nbsp;</span></p>

<p style='margin-left:36.0pt'><span lang=EN-US><!--[if gte vml 1]><v:shape
 id="_x0000_i1050" type="#_x0000_t75" alt="" style='width:208.5pt;height:21pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image025.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_268.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=278 height=28
src="./第六章%20附录——图形变换.files/image025.gif" v:shapes="_x0000_i1050"><![endif]></span></p>

<p><span lang=EN-US>&nbsp; 6) 矩阵的逆 </span></p>

<p style='margin-left:36.0pt;line-height:200%'>对于一个<i><span lang=EN-US
style='font-family:"Times New Roman"'>n</span></i><span lang=EN-US>x</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>n</span></i>的方阵<i><span
lang=EN-US style='font-family:"Times New Roman"'>A</span></i>，果存在一个<i><span
lang=EN-US style='font-family:"Times New Roman"'>n</span></i><span lang=EN-US>x</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>n</span></i>的方阵<i><span
lang=EN-US style='font-family:"Times New Roman"'>B</span></i>，使得<i><span
lang=EN-US style='font-family:"Times New Roman"'>AB</span></i><span lang=EN-US>=</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>BA</span></i><span lang=EN-US>=</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>I<sub>n</sub></span></i>，则称<span
lang=EN-US>B是A的<span style='color:maroon'>逆</span>，记为</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>B=A</span></i><sup><span
lang=EN-US style='font-family:"Times New Roman"'>-1</span></sup>，<span
lang=EN-US>A则被称为<span style='color:maroon'>非奇异矩阵</span>。</span></p>

<p style='margin-left:36.0pt;line-height:150%'>矩阵的逆是相互的，<i><span lang=EN-US
style='font-family:"Times New Roman"'>A</span></i>同样也可记为<i><span lang=EN-US
style='font-family:"Times New Roman"'>B=A</span></i><sup><span lang=EN-US
style='font-family:"Times New Roman"'>-1</span></sup>，<i><span lang=EN-US
style='font-family:"Times New Roman"'>B</span></i>也是一个非奇异矩阵。</p>

<p style='margin-left:36.0pt;line-height:150%'>任何非奇异矩阵有且只有一个逆矩阵。</p>

<p><span lang=EN-US>&nbsp; 7) 矩阵运算的基本性质 </span></p>

<ol start=1 type=A>
 <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
     mso-list:l21 level1 lfo16;tab-stops:list 36.0pt'><span style='font-family:
     宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>矩阵加法适合交换律与结合律</span><span
     lang=EN-US> </span></li>
</ol>

<p align=center style='margin-left:36.0pt;text-align:center'><span lang=EN-US><!--[if gte vml 1]><v:shape
 id="_x0000_i1051" type="#_x0000_t75" alt="" style='width:119.25pt;height:38.25pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image026.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_269.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=159 height=51
src="./第六章%20附录——图形变换.files/image026.gif" v:shapes="_x0000_i1051"><![endif]></span></p>

<ol start=2 type=A>
 <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
     mso-list:l21 level1 lfo16;tab-stops:list 36.0pt'><span style='font-family:
     宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>数乘矩阵适合分配律与结合律</span><span
     lang=EN-US> </span></li>
</ol>

<p align=center style='margin-left:36.0pt;text-align:center'><span lang=EN-US><!--[if gte vml 1]><v:shape
 id="_x0000_i1052" type="#_x0000_t75" alt="" style='width:139.5pt;height:38.25pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image027.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_270.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=186 height=51
src="./第六章%20附录——图形变换.files/image027.gif" v:shapes="_x0000_i1052"><![endif]></span></p>

<ol start=3 type=A>
 <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
     mso-list:l21 level1 lfo16;tab-stops:list 36.0pt'><span style='font-family:
     宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>矩阵的乘法适合结合律</span><span
     lang=EN-US> </span></li>
</ol>

<p align=center style='margin-left:36.0pt;text-align:center'><span lang=EN-US><!--[if gte vml 1]><v:shape
 id="_x0000_i1053" type="#_x0000_t75" alt="" style='width:87pt;height:21pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image028.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_271.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=116 height=28
src="./第六章%20附录——图形变换.files/image028.gif" v:shapes="_x0000_i1053"><![endif]></span></p>

<ol start=4 type=A>
 <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
     mso-list:l21 level1 lfo16;tab-stops:list 36.0pt'><span style='font-family:
     宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>矩阵的乘法对加法适合分配律</span><span
     lang=EN-US> </span></li>
</ol>

<p align=center style='margin-left:36.0pt;text-align:center'><span lang=EN-US><!--[if gte vml 1]><v:shape
 id="_x0000_i1054" type="#_x0000_t75" alt="" style='width:102.75pt;height:38.25pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image029.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_272.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=137 height=51
src="./第六章%20附录——图形变换.files/image029.gif" v:shapes="_x0000_i1054"><![endif]></span></p>

<ol start=5 type=A>
 <li class=MsoNormal style='mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;
     mso-list:l21 level1 lfo16;tab-stops:list 36.0pt'><span style='font-family:
     宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>矩阵的乘法不适合交换率</span><span
     lang=EN-US> </span></li>
</ol>

<p align=center style='margin-right:36.0pt;margin-left:36.0pt;text-align:center'><span
lang=EN-US><!--[if gte vml 1]><v:shape id="_x0000_i1055" type="#_x0000_t75"
 alt="" style='width:55.5pt;height:15.75pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image030.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_273.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=74 height=21
src="./第六章%20附录——图形变换.files/image030.gif" v:shapes="_x0000_i1055"><![endif]></span></p>

<p><span lang=EN-US><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>

<p><b><span lang=EN-US style='font-size:10.0pt;font-family:幼圆;color:gray'><a
href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Txt_6_007.htm"><span
style='text-decoration:none;text-underline:none'><!--[if gte vml 1]><v:shape
 id="_x0000_i1056" type="#_x0000_t75" alt="" href="..\CG_Txt_6_007.htm"
 style='width:21pt;height:37.5pt' o:button="t">
 <v:imagedata src="./第六章%20附录——图形变换.files/image001.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/material/CG_Gif_pub_021.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=28 height=50
src="./第六章%20附录——图形变换.files/image001.gif" v:shapes="_x0000_i1056"><![endif]></span></a></span></b><b><span
lang=EN-US style='font-family:幼圆;color:gray'><a
href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Txt_6_007.htm"><span
style='color:gray'>齐次坐标</span></a><o:p></o:p></span></b></p>

<p style='line-height:200%'>所谓<span style='color:maroon'>齐次坐标</span>就是将一个原本是<i><span
lang=EN-US style='font-family:"Times New Roman"'>n</span></i>维的向量用一个<i><span
lang=EN-US style='font-family:"Times New Roman"'>n</span></i><span lang=EN-US>+1维向量来表示。<!--[if gte vml 1]><v:shape
 id="_x0000_i1057" type="#_x0000_t75" alt="" style='width:403.5pt;height:21.75pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image031.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_277.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=538 height=29
src="./第六章%20附录——图形变换.files/image031.gif" v:shapes="_x0000_i1057"><![endif]> 实数。显然一个向量的齐次表示是不唯一的，齐次坐标的</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>h</span></i>取不同的值都表示的是同一个点，比如齐次坐标<span
lang=EN-US>[8,4,2]、[4,2,1]表示的都是二维点[2,1]。</span></p>

<p style='line-height:200%'>那么引进齐次坐标有什么必要，它有什么优点呢？</p>

<p style='margin-left:36.0pt;text-indent:-18.0pt;line-height:200%;mso-list:
l1 level1 lfo17;tab-stops:list 36.0pt'><![if !supportLists]><span lang=EN-US>A.<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span><![endif]>它提供了用矩阵运算把二维、三维甚至高维空间中的一个点集从一个坐标系变换到另一个坐标系的有效方法。</p>

<p style='margin-left:36.0pt;text-indent:-18.0pt;line-height:200%;mso-list:
l1 level1 lfo17;tab-stops:list 36.0pt'><![if !supportLists]><span lang=EN-US>B.<span
style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp; </span></span><![endif]>它可以表示无穷远的点。<i><span
lang=EN-US style='font-family:"Times New Roman"'>n</span></i><span lang=EN-US>+1维的齐次坐标中如果</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>h</span></i><span lang=EN-US>=0，实际上就表示了</span><i><span
lang=EN-US style='font-family:"Times New Roman"'>n</span></i>维空间的一个无穷远点。对于齐次坐标<span
lang=EN-US>[</span><i><span lang=EN-US style='font-family:"Times New Roman"'>a</span></i><span
lang=EN-US>,</span><i><span lang=EN-US style='font-family:"Times New Roman"'>b</span></i><span
lang=EN-US>,</span><i><span lang=EN-US style='font-family:"Times New Roman"'>h</span></i><span
lang=EN-US>]，保持</span><i><span lang=EN-US style='font-family:"Times New Roman"'>a</span></i><span
lang=EN-US>,</span><i><span lang=EN-US style='font-family:"Times New Roman"'>b</span></i>不变，<span
lang=EN-US><!--[if gte vml 1]><v:shape id="_x0000_i1058" type="#_x0000_t75"
 alt="" style='width:339pt;height:25.5pt'>
 <v:imagedata src="./第六章%20附录——图形变换.files/image032.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Gif_6_278.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=452 height=34
src="./第六章%20附录——图形变换.files/image032.gif" v:shapes="_x0000_i1058"><![endif]>&nbsp;&nbsp;&nbsp;
点沿直线 </span><i><span lang=EN-US style='font-family:"Times New Roman"'>ax</span></i><span
lang=EN-US>+</span><i><span lang=EN-US style='font-family:"Times New Roman"'>by</span></i><span
lang=EN-US>=0 逐渐走向无穷远处的过程。</span></p>

<p><span lang=EN-US><![if !supportEmptyParas]>&nbsp;<![endif]><o:p></o:p></span></p>

<p><b><span lang=EN-US style='font-size:10.0pt;font-family:幼圆;color:gray'><a
href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Txt_6_008.htm"><span
style='text-decoration:none;text-underline:none'><!--[if gte vml 1]><v:shape
 id="_x0000_i1059" type="#_x0000_t75" alt="" href="..\CG_Txt_6_008.htm"
 style='width:21pt;height:37.5pt' o:button="t">
 <v:imagedata src="./第六章%20附录——图形变换.files/image001.gif" o:href="http://learn.bitsde.com/hep/jisuanjituxing/material/CG_Gif_pub_021.gif"/>
</v:shape><![endif]--><![if !vml]><img border=0 width=28 height=50
src="./第六章%20附录——图形变换.files/image001.gif" v:shapes="_x0000_i1059"><![endif]></span></a></span></b><b><span
lang=EN-US style='font-family:幼圆;color:gray'><a
href="http://learn.bitsde.com/hep/jisuanjituxing/Chapter6/CG_Txt_6_008.htm"><span
style='color:gray'>线性方程组的求解</span></a></span></b></p>

<p>对于一个一个有<span lang=EN-US>n个变量的方程组：</span></p>

<p align=center style='text-align:center'><span lang=EN-US><!--[if gte vml 1]><v:shape
 id="_x0000_i1060" type="#_x0000_t75" alt="" style='width:189.75pt;height:86.25pt'>
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