线性代数英文讲义943418949.doc

Chapter 3-Section 1 Examples and Definition羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿蒅螂羄莈薇羈羀莇螀螀艿莇葿肆膅莆薁衿肁莅蚄肄羇莄螆袇芆蒃蒆蚀膂蒂薈袅肈蒁蚀蚈肄蒁蒀羄羀蒀薂螆芈葿蚅羂膄蒈螇螅肀蒇蒇羀羆薆蕿螃芅薅蚁羈膁薅袄螁膇薄薃肇肃膀蚆袀罿腿螈肅芇腿蒇袈膃膈薀肃聿芇蚂袆羅芆螄虿芄芅蒄袄芀芄蚆蚇膆芃蝿羃肂芃蒈螆羈节薁羁芇芁蚃螄膃莀螅罿肈荿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Chapter 3 Vector SpacesThe operations of addition and scalar multiplication are used in many diverse contexts in mathematics. Regardless of the context, however, these operations usually obey the same set of algebraic rules. Thus a general theory of mathematical systems involving addition and scalar multiplication will have applications to many areas in mathematics. 1. Examples and DefinitionNew words and phrasesVector space 向量空间Polynomial 多项式Degree 次数Axiom 公理Additive inverse 加法逆1.1 Examples Examining the following sets:(1) V=: The set of all vectors (2) V=: The set of all mxn matrices (3) V=: The set of all continuous functions on the interval (4) V=: The set of all polynomials of degree less than n. Question 1: What do they have in common? We can see that each of the sets, there are two operations: addition and multiplication, i.e. with each pair of elements x and y in a set V, we can associate a unique element x+y that is also an element in V, and with each element x and each scalar , we can associate a unique element in V. And the operations satisfy some algebraic rules. More generally, we introduce the concept of vector space. .1.2 Vector Space AxiomsDefinition Let V be a set on which the operations of addition and scalar multiplication are defined. By this we mean that, with each pair of elements x and y in a set V, we can associate a unique element x+y that is also an element in V, and with each element x and each scalar , we can associate a unique element in V. The set V together with the operations of addition and scalar multiplication is said to form a vector space if the following axioms are satisfied.A1. x+y=y+x for any x and y in V.A2. (x+y)+z=x+(y+z) for any x, y, z in V.A3. There exists an element 0 in V such that x+0=x for each x in V.A4. For each x in V, there exists an element x in V such that x+(-x)=0.A5. (x+y)= x+y for each scalar and any x and y in V.A6. (+)x=x+x for any scalars and and any x in V.A7. ()x=(x) for any scalars and and any x in V.A8. 1x=x for all x in V.From this definition, we see that the examples in 1.1 are all vector spaces.In the definition, there is an important component, the closure properties of the two operations. These properties are summarized as follows:C1. If x is in V and is a scalar, then x is in VC2. If x, y are in V, then x+y is in V.An example that is not a vector space:Let , on this set, the addition and multiplication are defined in the usually way. The operation + and scalar multiplication are not defined on W. The sum of two vector is not necessarily in W, neither is the scalar multiplication. Hence, W together with the addition and multiplication is not a vector space. In the examples in 1.1, we see that the following statements are true.Theorem 3.1.1 If V is a vector space and x is any element of V, then (i) 0x=0 (ii) x+y=0 implies that y=-x (i.e. the additive inverse is unique).(iii) (-1)x=-x.But is this true for any vector space?Question: Are they obvious? Do we have to prove them? But if we look at the definition of vector space, we dont know what the elements are, how the addition and multiplication are defined. So theorem above is not very obvious.Proof (i) x=1x=(1+0)x=1x+0x=x+0x, (A6 and A8)Thus x+x=-x+(x+0x)=(-x+x)+0x (A2) 0=0+0x=0x (A1, A3, and A4)(ii) Suppose that x+y=0. then -x=-x+0=-x+(x+y)Therefore, -x=(-x+x)+y=0+y=y(iii) 0=0x=(1+(-1)x=1x+(-1)x, thus x+(-1)x=0 It follows from part (ii) that (-1)x=-xAssignment for section 1, chapter 3Hand in: 9, 10, 12.Chapter 3-Section 2 Subspaces2. SubspacesNew words and phrasesSubspace 子空间Trivial subspace 平凡子空间Proper subspace 真子空间Span 生成Spanning set生成集Nullspace 零空间2.1 Definition Given a vector space V, it is often possible to form another vector space by taking a subset of V and using the operations of V. For a new subset S of V to be a vector space, the set S must be closed under the operations of addition and scalar multiplication. Examples (on page 124)The set together with the usual addition and scalar multiplication is itself a vector space .The set S=together with the usual addition and scalar multiplication is itself a vector space. Definition If S is a nonempty subset of a vector space V, and S satisfies the following conditions: (i) xS whenever xS for any scalar (ii) x+y S whenever xS and ySthen S is said to be a subspace （子空间）of V.A subspace S of V together with the operations of addition and scalar multiplication satisfies all the conditions in the definition of a vector space. Hence, every subspace of a vector space is a vector space in its own right. Trivial Subspaces and Proper SubspacesThe set containing only the zero element forms a subspace, called zero subspace, and V is also a subspace of V. Those two subspaces are called trivial subspaces of V. All other subspaces are referred to as proper subspaces. Examples of Subspaces (1) the set of all differentiable functions on a,b is a subspace of (2) the set of all polynomials of degree less than n (1) with the property p(0) form a subspace of . (3) the set of matrices of the form forms a subspace of . (4) the set of all mxm symmetric matrices forms a subspace of (5) the set of all mxm skew-symmetric matrices form a subspace of 2.2 The Nullspace of a Matrix Let A be an mxn matrix, and .Then N(A) form a subspace of . The subspace N(A) is called the nullspace of A. The proof is a straightforward verification of the definition.2.3 The Span of a Set of VectorsIn this part, we give a method for forming a subspace of V with finite number of vectors in V.Given n vectors in a vector space of V, we can form a new subset of V as the following.It is easy to show that this set forms a subset of V. We call this subspace the span of , or the subspace of V spanned by .Theorem 3.2.1 If are elements of a vector space of V, then is a subspace of V.For example, the subspace spanned by two vectors and is the subspace consisting of the elements .2.4 Spanning Set for a Vector SpaceDefinition If are vectors of V and V=, then the set is called a spanning set （生成集）for V. In other words, the set is a spanning set for V if and only if every element can be written as a linear combination of .The spanning sets for a vector space are not unique. Examples (Determining if a set spans for )(a) (b) (c) (d) To do this, we have to show that every vector in can be written as a linear combination of the given vectors. Assignment for section 2, chapter 3Hand in: 6, 8, 13, 16, 17, 18, 20Not required: 21Chapter 3-Section 3 Linear Independence3. Linear IndependenceNew words and phrasesLinear independence 线性无关性Linearly independent 线性无关的Linear dependence 线性相关性Linearly dependent 线性相关的3.1 Motivation In this section, we look more closely at the structure of vector spaces. We restrict ourselves to vector spaces that can be generated from a finite set of elements, or vector spaces that are spans of finite number of vectors. V=The set is called a generating set or spanning set(生成集). It is desirable to find a minimal spanning set. By minimal, we mean a spanning set with no unnecessary element. To see how to find a minimal spanning set, it is necessary to consider how the vectors in the collection depend on each other. Consequently we introduce the concepts of linear dependence and linear independence. These simple concepts provide the keys to understanding the structure of vector spaces. Give an example in which we can reduce the number of vectors in a spanning set. Consider the following three vectors in . These three vectors satisfy (1) Any linear combination of can be reduced to a linear combination of . Thus S= Span()=Span(). (2) (a dependency relation)Since the three coefficients are nonzero, we could solve for any vector in terms of the other two. It follows that Span()=Span()=Span()=Span()On the other hand, no such dependency relationship exists between . In deed, if there were scalars and , not both 0, such that (3) then we could solve for one of the two vectors in terms of the other. However, neither of the two vectors in question is a multiple of the other. Therefore, Span() and Span() are both proper subspaces of Span(), and the only way that (3) can hold is if .Observations:(I) If span a vector space V and one of these vectors can be written as a linear combination of the other n-1 vectors, then those n-1 vectors span V.(II) Given n vectors , it is possible to write one of the vectors as a linear combination of the other n-1 vectors if and only if there exist scalars not all zero such that Proof of I: Suppose that can be written as a linear combination of the vectors . Proof of II: The key point here is that there at least one nonzero coefficient. 3.2 DefinitionsDefinition The vectors in a vector space V are said to be linearly independent（线性独立的） if implies that all the scalars must equal zero. Example: are linearly independent. Definition The vectors in a vector space V are said to be linearly dependent （线性相关的）if there exist scalars not all zero such that.Let be vector in . Then are linearly dependent. If there are nontrivial choices of scalars for which the linear combination equals the zero vector, then are linearly dependent. If the only way the linear combination can equal the zero vector is for all scalars to be 0, then are linearly independent.3.3 Geometric Interpretation The linear dependence and independence in and . Each vector in or represents a directed line segment originated at the origin. Two vector are linearly dependent in or if and only if two vectors are collinear. Three or more vector in must be linearly dependent. Three vectors in are linearly dependent if and only if three vectors are coplanar. Four or more vectors in must be linearly dependent. 3.4 Theorems and ExamplesIn this part, we learn some theorems that tell whether a set of vectors is linearly independent. Example: (Example 3 on page 138) Which of the following collections of vectors are linearly independent?(a) (b) (c) (d) The problem of determining the linear dependency of a collection of vectors in can be reduced to a problem of solving a linear homogeneous system. If the system has only the trivial solution, then the vectors are linearly independent, otherwise, they are linearly dependent, We summarize the this method in the following theorem: Theorem n vectors in are linearly dependent if the linear system Xc=0 has a nontrivial solution, where .Proof: Xc=0.Theorem 3.3.1 Let be n vec