第六章线性空间习题提高题/title>  <style>  </style>  </head> <body lang=ZH-CN style='tab-interval:21.0pt;text-justify-trim:punctuation'> <div class=Section1 style='layout-grid:15.6pt'> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span lang=EN-US><span style='mso-spacerun:yes'> </span></span></b><b style='mso-bidi-font-weight:normal'><span style='font-size:14.0pt;font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>第六章</span></b><b style='mso-bidi-font-weight: normal'><span lang=EN-US style='font-size:14.0pt'><span style='mso-spacerun:yes'> </span></span></b><b style='mso-bidi-font-weight: normal'><span style='font-size:14.0pt;font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>线性空间习题</span></b><b style='mso-bidi-font-weight:normal'><span lang=EN-US style='font-size:14.0pt'><o:p></o:p></span></b></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span style='font-size:14.0pt;font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>提高题</span><span lang=EN-US><o:p></o:p></span></b></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span style='font-size:12.0pt;font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>一．填空、计算题</span></b><b style='mso-bidi-font-weight:normal'><span lang=EN-US style='font-size:12.0pt'><o:p></o:p></span></b></p> <p class=MsoNormal style='text-indent:21.0pt;mso-char-indent-count:2.0'><span lang=EN-US>1.</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>（孝感大学</span><span lang=EN-US>2004</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>年考研）设列向量组</span><span lang=EN-US><sub><![if !vml]><img width=79 height=24 src="diliuzhangT.files/image002.gif" v:shapes="_x0000_i1025"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>与</span><span lang=EN-US><sub><![if !vml]><img width=79 height=24 src="diliuzhangT.files/image004.gif" v:shapes="_x0000_i1026"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>是</span><span lang=EN-US><sub><![if !vml]><img width=21 height=20 src="diliuzhangT.files/image006.gif" v:shapes="_x0000_i1027"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>的一组基，记矩阵</span><span lang=EN-US><sub><![if !vml]><img width=119 height=24 src="diliuzhangT.files/image008.gif" v:shapes="_x0000_i1028"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>，</span><span lang=EN-US><sub><![if !vml]><img width=119 height=24 src="diliuzhangT.files/image010.gif" v:shapes="_x0000_i1029"><![endif]></sub> </span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>，则基</span><span lang=EN-US><sub><![if !vml]><img width=79 height=24 src="diliuzhangT.files/image002.gif" v:shapes="_x0000_i1030"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>到</span><span lang=EN-US><sub><![if !vml]><img width=79 height=24 src="diliuzhangT.files/image004.gif" v:shapes="_x0000_i1031"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>的过渡矩阵是</span><u><span lang=EN-US style='mso-bidi-font-size:10.5pt'><span style='mso-spacerun:yes'> </span></span></u><span lang=EN-US style='mso-bidi-font-size:10.5pt'>.</span></p> <p class=MsoNormal style='text-indent:23.5pt;mso-char-indent-count:1.96'><span lang=EN-US style='font-size:12.0pt'>2</span><span lang=EN-US>.</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>（武汉理工大学</span><span lang=EN-US>2001</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>年考研）设</span><span lang=EN-US><sub><![if !vml]><img width=91 height=24 src="diliuzhangT.files/image012.gif" v:shapes="_x0000_i1032"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>表示数域</span><i style='mso-bidi-font-style:normal'><span lang=EN-US style='font-size:12.0pt'>P</span></i><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>上的</span><span lang=EN-US>n</span><span style='font-family: 宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>阶方阵的全体。</span></p> <p class=MsoNormal><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>（</span><span lang=EN-US>1</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>）证明：</span><span lang=EN-US><sub><![if !vml]><img width=185 height=24 src="diliuzhangT.files/image014.gif" v:shapes="_x0000_i1033"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>是</span><span lang=EN-US><sub><![if !vml]><img width=31 height=20 src="diliuzhangT.files/image016.gif" v:shapes="_x0000_i1034"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>上的一个子空间。</span></p> <p class=MsoNormal><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>（</span><span lang=EN-US>2</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>）当</span><i style='mso-bidi-font-style:normal'><span lang=EN-US style='font-size:12.0pt'>A</span></i><span style='font-family:宋体; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>为对角阵，</span><span lang=EN-US><sub><![if !vml]><img width=111 height=24 src="diliuzhangT.files/image018.gif" v:shapes="_x0000_i1035"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>，求</span><i style='mso-bidi-font-style:normal'><span lang=EN-US style='font-size:12.0pt'>C</span></i><span lang=EN-US>(</span><i style='mso-bidi-font-style:normal'><span lang=EN-US style='font-size:12.0pt'>A</span></i><span lang=EN-US>)</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>的维数和一组基。</span></p> <p class=MsoNormal align=left style='text-align:left;text-indent:21.0pt; mso-char-indent-count:2.0;line-height:150%'><span lang=EN-US style='mso-bidi-font-size: 10.5pt;line-height:150%'>3.<span style='mso-bidi-font-weight:bold'> (1) </span></span><span style='mso-bidi-font-size:10.5pt;line-height:150%;font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman";mso-bidi-font-weight: bold'>证明：在</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height: 150%;mso-bidi-font-weight:bold'><sub><![if !vml]><img width=40 height=24 src="diliuzhangT.files/image020.gif" v:shapes="_x0000_i1036"><![endif]></sub></span><span style='mso-bidi-font-size:10.5pt;line-height: 150%;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman";mso-bidi-font-weight:bold'>中，多项式</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%;mso-bidi-font-weight:bold'><sub><![if !vml]><img width=280 height=24 src="diliuzhangT.files/image022.gif" v:shapes="_x0000_i1037"><![endif]></sub></span><span style='mso-bidi-font-size:10.5pt;line-height: 150%;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman";mso-bidi-font-weight:bold'>，</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%;mso-bidi-font-weight:bold'><sub><![if !vml]><img width=79 height=21 src="diliuzhangT.files/image024.gif" v:shapes="_x0000_i1038"><![endif]></sub></span><span style='mso-bidi-font-size:10.5pt;line-height: 150%;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman";mso-bidi-font-weight:bold'>，是一组基，其中</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%;mso-bidi-font-weight:bold'><sub><![if !vml]><img width=81 height=24 src="diliuzhangT.files/image026.gif" v:shapes="_x0000_i1039"><![endif]></sub></span><span style='mso-bidi-font-size:10.5pt;line-height: 150%;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman";mso-bidi-font-weight:bold'>是互不相同的数；</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%;mso-bidi-font-weight:bold'><o:p></o:p></span></p> <p class=MsoNormal align=left style='text-align:left;line-height:150%'><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%;mso-bidi-font-weight: bold'><span style='mso-spacerun:yes'> </span>(2)</span><span style='mso-bidi-font-size:10.5pt;line-height:150%;font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman";mso-bidi-font-weight: bold'>在</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height: 150%;mso-bidi-font-weight:bold'>1</span><span style='mso-bidi-font-size:10.5pt; line-height:150%;font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-bidi-font-weight:bold'>）中，取</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%;mso-bidi-font-weight: bold'><sub><![if !vml]><img width=81 height=24 src="diliuzhangT.files/image026.gif" v:shapes="_x0000_i1040"><![endif]></sub></span><span style='mso-bidi-font-size:10.5pt;line-height: 150%;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman";mso-bidi-font-weight:bold'>是全体</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%;mso-bidi-font-weight:bold'>n</span><span style='mso-bidi-font-size:10.5pt;line-height:150%;font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman";mso-bidi-font-weight: bold'>次单位根，求由基</span><span lang=EN-US style='mso-bidi-font-size:10.5pt; line-height:150%;mso-bidi-font-weight:bold'><sub><![if !vml]><img width=75 height=24 src="diliuzhangT.files/image028.gif" v:shapes="_x0000_i1041"><![endif]></sub></span><span style='mso-bidi-font-size:10.5pt;line-height: 150%;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman";mso-bidi-font-weight:bold'>到基</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%;mso-bidi-font-weight:bold'><sub><![if !vml]><img width=81 height=24 src="diliuzhangT.files/image030.gif" v:shapes="_x0000_i1042"><![endif]></sub></span><span style='mso-bidi-font-size:10.5pt;line-height: 150%;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman";mso-bidi-font-weight:bold'>的过渡矩阵。</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%;mso-bidi-font-weight:bold'><o:p></o:p></span></p> <p class=MsoNormal style='text-indent:21.75pt'><span lang=EN-US>4.</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>（北京大学</span><span lang=EN-US>2005</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>年考研）设数域</span><span lang=EN-US>K</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>上的</span><span lang=EN-US>n</span><span style='font-family: 宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>级矩阵</span><span lang=EN-US>A</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>的（</span><i style='mso-bidi-font-style: normal'><span lang=EN-US>i,j</span></i><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>）元为</span><span lang=EN-US><sub><![if !vml]><img width=43 height=25 src="diliuzhangT.files/image032.gif" v:shapes="_x0000_i1043"><![endif]></sub></span></p> <p class=MsoNormal style='text-indent:21.75pt'><span style='font-family:宋体; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>（</span><span lang=EN-US>1</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>）求</span><span lang=EN-US>|A|.</span></p> <p class=MsoNormal style='text-indent:21.75pt'><span style='font-family:宋体; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>（</span><span lang=EN-US>2</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>）当</span><span lang=EN-US><sub><![if !vml]><img width=37 height=19 src="diliuzhangT.files/image034.gif" v:shapes="_x0000_i1044"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>时，</span><span lang=EN-US><sub><![if !vml]><img width=93 height=24 src="diliuzhangT.files/image036.gif" v:shapes="_x0000_i1045"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>，求齐次线性方程组</span><span lang=EN-US><sub><![if !vml]><img width=56 height=19 src="diliuzhangT.files/image038.gif" v:shapes="_x0000_i1046"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>的维数与一组基。</span></p> <p class=MsoNormal style='text-indent:21.75pt'><span lang=EN-US>5</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>．</span><span lang=EN-US>(</span><span style='font-family: 宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>北京大学</span><span lang=EN-US>2005</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>年考研）用</span><span lang=EN-US><sub><![if !vml]><img width=51 height=24 src="diliuzhangT.files/image040.gif" v:shapes="_x0000_i1047"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>表示数域</span><i style='mso-bidi-font-style:normal'><span lang=EN-US style='font-size:12.0pt'>K</span></i><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>中所有</span><span lang=EN-US>n</span><span style='font-family: 宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>级矩阵组成的集合，它对于矩阵的加法与乘法成为</span><i style='mso-bidi-font-style:normal'><span lang=EN-US style='font-size:12.0pt'>K</span></i><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>上的线性空间。数域</span><i style='mso-bidi-font-style:normal'><span lang=EN-US style='font-size:12.0pt'>K</span></i><span style='font-family:宋体; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>上的</span><span lang=EN-US> n</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>级矩阵</span> </p> <p class=MsoNormal style='text-indent:21.75pt'><span lang=EN-US><span style='mso-spacerun:yes'> </span><sub><![if !vml]><img width=188 height=123 src="diliuzhangT.files/image042.gif" v:shapes="_x0000_i1048"><![endif]></sub></span></p> <p class=MsoNormal><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>称为循环矩阵，</span><i style='mso-bidi-font-style: normal'><span lang=EN-US style='font-size:12.0pt'>U</span></i><span style='mso-bidi-font-size:10.5pt;font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>表示</span><span style='font-family: 宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>数域</span><i style='mso-bidi-font-style:normal'><span lang=EN-US style='font-size:12.0pt'>K</span></i><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>上所有</span><span lang=EN-US>n</span><span style='font-family: 宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>级矩阵的集合。证明：</span><i style='mso-bidi-font-style:normal'><span lang=EN-US style='font-size:12.0pt'>U</span></i><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>是</span><span lang=EN-US><sub><![if !vml]><img width=51 height=24 src="diliuzhangT.files/image040.gif" v:shapes="_x0000_i1049"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>的子空间，并求</span><i style='mso-bidi-font-style:normal'><span lang=EN-US style='font-size:12.0pt'>U</span></i><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>的基和维数。</span></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span lang=EN-US style='font-size:12.0pt'><o:p> </o:p></span></b></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span style='font-size:12.0pt;font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>二</span></b><b style='mso-bidi-font-weight: normal'><span lang=EN-US style='font-size:12.0pt'>.</span></b><b style='mso-bidi-font-weight:normal'><span style='font-size:12.0pt;font-family: 宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>证明题</span></b><b style='mso-bidi-font-weight:normal'><span lang=EN-US style='font-size:12.0pt'><o:p></o:p></span></b></p> <p class=MsoNormal style='text-indent:20.6pt;mso-char-indent-count:1.96'><span lang=EN-US>1. </span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>（复旦大学</span><span lang=EN-US>2000</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>年考研）设</span><span lang=EN-US><sub><![if !vml]><img width=13 height=15 src="diliuzhangT.files/image044.gif" v:shapes="_x0000_i1050"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>是一个自然数，</span><span lang=EN-US><sub><![if !vml]><img width=16 height=19 src="diliuzhangT.files/image046.gif" v:shapes="_x0000_i1051"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>是由所有</span><span lang=EN-US><sub><![if !vml]><img width=35 height=15 src="diliuzhangT.files/image048.gif" v:shapes="_x0000_i1052"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>实矩阵构成的</span><span lang=EN-US><sub><![if !vml]><img width=19 height=21 src="diliuzhangT.files/image050.gif" v:shapes="_x0000_i1053"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>维实向量空间，</span><span lang=EN-US><sub><![if !vml]><img width=17 height=19 src="diliuzhangT.files/image052.gif" v:shapes="_x0000_i1054"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>和</span><span lang=EN-US><sub><![if !vml]><img width=19 height=19 src="diliuzhangT.files/image054.gif" v:shapes="_x0000_i1055"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>分别为由所有</span><span lang=EN-US><sub><![if !vml]><img width=35 height=15 src="diliuzhangT.files/image048.gif" v:shapes="_x0000_i1056"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>对称矩阵和反对称矩阵构成的空间。证明</span><span lang=EN-US><sub><![if !vml]><img width=77 height=19 src="diliuzhangT.files/image057.gif" v:shapes="_x0000_i1057"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>，既</span><span lang=EN-US><sub><![if !vml]><img width=16 height=19 src="diliuzhangT.files/image046.gif" v:shapes="_x0000_i1058"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>是</span><span lang=EN-US><sub><![if !vml]><img width=17 height=19 src="diliuzhangT.files/image052.gif" v:shapes="_x0000_i1059"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>和</span><span lang=EN-US><sub><![if !vml]><img width=19 height=19 src="diliuzhangT.files/image054.gif" v:shapes="_x0000_i1060"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>的直和。</span></p> <p class=MsoNormal style='text-indent:21.0pt;mso-char-indent-count:2.0'><span lang=EN-US>2.<sub><![if !vml]><img width=31 height=20 src="diliuzhangT.files/image062.gif" v:shapes="_x0000_i1061"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>表示数域</span><i style='mso-bidi-font-style:normal'><span lang=EN-US style='font-size:12.0pt'>P</span></i><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>上的</span><span lang=EN-US>2</span><span style='font-family: 宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"'>阶方阵的全体，设</span><span lang=EN-US><sub><![if !vml]><img width=76 height=24 src="diliuzhangT.files/image064.gif" v:shapes="_x0000_i1062"><![endif]></sub></span><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>为两两互异的数且和不为零。试证明</span></p> <p class=MsoNormal style='text-indent:52.5pt;mso-char-indent-count:5.0'><span lang=EN-US><sub><![if !vml]><img width=383 height=51 src="diliuzhangT.files/image066.gif" v:shapes="_x0000_i1063"><![endif]></sub></span></p> <p class=MsoNormal><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>为线性空间</span><b style='mso-bidi-font-weight: normal'><span lang=EN-US><sub><![if !vml]><img width=31 height=20 src="diliuzhangT.files/image062.gif" v:shapes="_x0000_i1064"><![endif]></sub></span></b><span style='font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>的基。</span><b style='mso-bidi-font-weight:normal'><span lang=EN-US><o:p></o:p></span></b></p> <p class=MsoNormal style='text-indent:21.0pt;mso-char-indent-count:2.0'><span lang=EN-US>3.</span><span lang=EN-US style='font-family:宋体'> (</span><span style='font-family:宋体'>华中科大<span lang=EN-US>2005</span>年考研<span lang=EN-US>)</span>设<span lang=EN-US><sub><![if !vml]><img width=63 height=20 src="diliuzhangT.files/image068.gif" v:shapes="_x0000_i1065"><![endif]></sub>,<sub><![if !vml]><img width=115 height=21 src="diliuzhangT.files/image070.gif" v:shapes="_x0000_i1066"><![endif]></sub>, </span>且<span lang=EN-US><sub><![if !vml]><img width=103 height=21 src="diliuzhangT.files/image072.gif" v:shapes="_x0000_i1067"><![endif]></sub>. </span>令<span lang=EN-US><sub><![if !vml]><img width=71 height=21 src="diliuzhangT.files/image074.gif" v:shapes="_x0000_i1068"><![endif]></sub>,<sub><![if !vml]><img width=69 height=21 src="diliuzhangT.files/image076.gif" v:shapes="_x0000_i1069"><![endif]></sub>, <sub><![if !vml]><img width=63 height=24 src="diliuzhangT.files/image078.gif" v:shapes="_x0000_i1070"><![endif]></sub><span style='mso-spacerun:yes'> </span></span>分别为线性方程组<span lang=EN-US><sub><![if !vml]><img width=65 height=19 src="diliuzhangT.files/image080.gif" v:shapes="_x0000_i1071"><![endif]></sub>,<sub><![if !vml]><img width=56 height=19 src="diliuzhangT.files/image038.gif" v:shapes="_x0000_i1072"><![endif]></sub>,<sub><![if !vml]><img width=55 height=19 src="diliuzhangT.files/image083.gif" v:shapes="_x0000_i1073"><![endif]></sub></span>的解空间<span lang=EN-US>.</span>证明<span lang=EN-US><sub><![if !vml]><img width=84 height=24 src="diliuzhangT.files/image085.gif" v:shapes="_x0000_i1074"><![endif]></sub>.</span></span></p> <p class=MsoNormal align=left style='text-align:left;text-indent:21.0pt; mso-char-indent-count:2.0;line-height:150%'><span lang=EN-US style='mso-bidi-font-size: 10.5pt;line-height:150%'>4.<span style='mso-bidi-font-weight:bold'> (1) </span></span><span style='mso-bidi-font-size:10.5pt;line-height:150%;font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman";mso-bidi-font-weight: bold'>证明：在</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height: 150%;mso-bidi-font-weight:bold'><sub><![if !vml]><img width=40 height=24 src="diliuzhangT.files/image020.gif" v:shapes="_x0000_i1075"><![endif]></sub></span><span style='mso-bidi-font-size:10.5pt;line-height: 150%;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman";mso-bidi-font-weight:bold'>中，多项</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%;mso-bidi-font-weight:bold'><sub><![if !vml]><img width=280 height=24 src="diliuzhangT.files/image022.gif" v:shapes="_x0000_i1076"><![endif]></sub></span><span style='mso-bidi-font-size:10.5pt;line-height: 150%;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman";mso-bidi-font-weight:bold'>，</span><i style='mso-bidi-font-style: normal'><span lang=EN-US style='font-size:12.0pt;line-height:150%;mso-bidi-font-weight: bold'>i</span></i><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height: 150%;mso-bidi-font-weight:bold'>=</span><span lang=EN-US style='font-size:12.0pt; line-height:150%;mso-bidi-font-weight:bold'>1<i style='mso-bidi-font-style: normal'>,…,n</i></span><i style='mso-bidi-font-style:normal'><span style='font-size:12.0pt;line-height:150%;font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman";mso-bidi-font-weight: bold'>，</span></i><span style='mso-bidi-font-size:10.5pt;line-height:150%; font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman";mso-bidi-font-weight:bold'>是一组基，其中</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%;mso-bidi-font-weight:bold'><sub><![if !vml]><img width=81 height=24 src="diliuzhangT.files/image026.gif" v:shapes="_x0000_i1077"><![endif]></sub></span><span style='mso-bidi-font-size:10.5pt;line-height: 150%;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman";mso-bidi-font-weight:bold'>是互不相同的数；</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%;mso-bidi-font-weight:bold'><o:p></o:p></span></p> <p class=MsoNormal align=left style='text-align:left;line-height:150%'><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%;mso-bidi-font-weight: bold'><span style='mso-spacerun:yes'> </span>(2)</span><span style='mso-bidi-font-size:10.5pt;line-height:150%;font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman";mso-bidi-font-weight: bold'>在</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height: 150%;mso-bidi-font-weight:bold'>1</span><span style='mso-bidi-font-size:10.5pt; line-height:150%;font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-bidi-font-weight:bold'>）中，取</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%;mso-bidi-font-weight: bold'><sub><![if !vml]><img width=81 height=24 src="diliuzhangT.files/image026.gif" v:shapes="_x0000_i1078"><![endif]></sub></span><span style='mso-bidi-font-size:10.5pt;line-height: 150%;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman";mso-bidi-font-weight:bold'>是全体</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%;mso-bidi-font-weight:bold'>n</span><span style='mso-bidi-font-size:10.5pt;line-height:150%;font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman";mso-bidi-font-weight: bold'>次单位根，求由基</span><span lang=EN-US style='mso-bidi-font-size:10.5pt; line-height:150%;mso-bidi-font-weight:bold'><sub><![if !vml]><img width=75 height=24 src="diliuzhangT.files/image028.gif" v:shapes="_x0000_i1079"><![endif]></sub></span><span style='mso-bidi-font-size:10.5pt;line-height: 150%;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman";mso-bidi-font-weight:bold'>到基</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%;mso-bidi-font-weight:bold'><sub><![if !vml]><img width=81 height=24 src="diliuzhangT.files/image030.gif" v:shapes="_x0000_i1080"><![endif]></sub></span><span style='mso-bidi-font-size:10.5pt;line-height: 150%;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman";mso-bidi-font-weight:bold'>的过渡矩阵。</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%;mso-bidi-font-weight:bold'><o:p></o:p></span></p> <p class=MsoNormal style='text-indent:21.0pt;mso-char-indent-count:2.0; line-height:150%'><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height: 150%'>5. </span><span style='mso-bidi-font-size:10.5pt;line-height:150%; font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>设</span><span lang=EN-US style='mso-bidi-font-size:10.5pt; line-height:150%'>F</span><span style='mso-bidi-font-size:10.5pt;line-height: 150%;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>为数域</span><span lang=EN-US style='mso-bidi-font-size:10.5pt; line-height:150%'>,<sub><![if !vml]><img width=33 height=21 src="diliuzhangT.files/image087.gif" v:shapes="_x0000_i1081"><![endif]></sub></span><span style='mso-bidi-font-size:10.5pt;line-height: 150%;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>为复数</span><span lang=EN-US style='mso-bidi-font-size:10.5pt; line-height:150%'>,<o:p></o:p></span></p> <p class=MsoNormal align=center style='text-align:center;line-height:150%'><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%'><sub><![if !vml]><img width=195 height=24 src="diliuzhangT.files/image089.gif" v:shapes="_x0000_i1082"><![endif]></sub><o:p></o:p></span></p> <p class=MsoNormal align=center style='text-align:center;line-height:150%'><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%'><sub><![if !vml]><img width=193 height=25 src="diliuzhangT.files/image091.gif" v:shapes="_x0000_i1083"><![endif]></sub><o:p></o:p></span></p> <p class=MsoNormal style='line-height:150%'><span style='mso-bidi-font-size: 10.5pt;line-height:150%;font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>证明</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%'>:</span><span style='mso-bidi-font-size:10.5pt;line-height:150%;font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>对于多项式的加法及数与多项式的乘法</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%'>,<sub><![if !vml]><img width=43 height=25 src="diliuzhangT.files/image093.gif" v:shapes="_x0000_i1084"><![endif]></sub></span><span style='mso-bidi-font-size:10.5pt;line-height: 150%;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>均作成</span><i style='mso-bidi-font-style:normal'><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%'>F</span></i><span style='mso-bidi-font-size:10.5pt;line-height:150%;font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>上向量空间</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%'>,</span><span style='mso-bidi-font-size:10.5pt;line-height:150%;font-family:宋体;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"'>且</span><span lang=EN-US style='mso-bidi-font-size:10.5pt;line-height:150%'><sub><![if !vml]><img width=53 height=25 src="diliuzhangT.files/image095.gif" v:shapes="_x0000_i1085"><![endif]></sub></span><span style='mso-bidi-font-size:10.5pt;line-height: 150%;font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>同构</span><span lang=EN-US style='mso-bidi-font-size:10.5pt; line-height:150%'>.<o:p></o:p></span></p> <p class=MsoNormal style='text-indent:21.75pt'><span lang=EN-US><o:p> </o:p></span></p> <p class=MsoNormal style='text-indent:21.75pt'><span lang=EN-US><o:p> </o:p></span></p> <p class=MsoNormal style='text-indent:21.75pt'><span lang=EN-US><o:p> </o:p></span></p> <p class=MsoNormal style='text-indent:273.0pt;mso-char-indent-count:26.0'><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>编辑：忠杰</span><span lang=EN-US><o:p></o:p></span></p> <p class=MsoNormal><span lang=EN-US><span style='mso-spacerun:yes'> </span></span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>网站：瀚海网：</span><span lang=EN-US>http://hanhai.org <o:p></o:p></span></p> <p class=MsoNormal><span lang=EN-US><span style='mso-spacerun:yes'> </span></span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>邮箱：</span><span lang=EN-US>nchy_zouzij@163.com<o:p></o:p></span></p> <p class=MsoNormal><span lang=EN-US><span style='mso-spacerun:yes'> </span>2009</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"'>．</span><span lang=EN-US>5</span><span style='font-family:宋体;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"'>．</span><span lang=EN-US>20<o:p></o:p></span></p> <p class=MsoNormal><span lang=EN-US style='font-family:宋体'><o:p> </o:p></span></p> <p class=MsoNormal style='text-indent:273.0pt;mso-char-indent-count:26.0'><span lang=EN-US><o:p> </o:p></span></p> </div> </body> </html>