《线性代数》（陈建龙等）第二章 n维向量.ppt

René Descartes Born: 31 March 1596 in La Haye (now Descartes),Touraine, France Died: 11 Feb 1650 in Stockholm, Sweden Pierre de Fermat Born: 17 Aug 1601 in Beaumont-de-Lomagne, France Died: 12 Jan 1665 in Castres, France Johann Carl Friedrich Gauss Born: 30 April 1777 in Brunswick, Duchy of Brunswick (now Germany) Died: 23 Feb 1855 in G?ttingen, Hanover (now Germany) Jorgen Pedersen Gram Born: 27 June 1850 in Nustrup (18 km W of Haderslev), Denmark Died: 29 April 1916 in Copenhagen, Denmark Hermann Günter Grassmann Born: 15 April 1809 in Stettin, Prussia (now Szczecin, Poland) Died: 26 Sept 1877 in Stettin, Germany (now Szczecin, Poland) Oliver Heaviside Born: 18 May 1850 in Camden Town, London, England Died: 3 Feb 1925 in Torquay, Devon, England Giuseppe Peano Born: 27 Aug 1858 in Cuneo, Piemonte, Italy Died: 20 April 1932 in Turin, Italy Erhard Schmidt Born: 13 Jan 1876 in Dorpat, Germany (now Tartu, Estonia) Died: 6 Dec 1959 in Berlin, Germany ? ? ? ? §2.3 向量组线性相关性的等价刻画 例7. 证明: n个n维列向量?1, ?2, …, ?n线性无 关的充分必要条件是: 任何一个n维列向 量?都能由?1, ?2, …, ?n线性表示. 证明: (充分性) 任何一个n维列向量? 都能由 ?1, ?2, …, ?n线性表示? ?1 = 1 0 0 … , ?2 = 0 1 0 … , …, ?n = 0 0 1 … 都能由?1, ?2, …, ?n线性表示? n = r(?1, …, ?n) ? r(?1, …, ?n) ? n ? … 第二章 n维列向量 ? §2.3 向量组线性相关性的等价刻画 证明: (必要性) 由于n+1个n维列向量总是线 性相关的, 所以?1, ?2, …, ?n, ?线性相 关. 又因为?1, ?2, …, ?n线性无关, 根据定理 2.4可知?都能由?1, ?2, …, ?n线性表示. 第二章 n维列向量 例7. 证明: n个n维列向量?1, ?2, …, ?n线性无 关的充分必要条件是: 任何一个n维列向 量?都能由?1, ?2, …, ?n线性表示. ? §2.4 向量组的极大线性无关组 第二章 n维列向量 §2.4 向量组的极大线性无关组 一. 定义 如果向量组?1, ?2, …, ?s的部分组 满足以下列条件: , …, ? i1 ? , ? i2 ir 线性无关; , …, ? i1 (1) ? , ? i2 ir (2) ?1, ?2, …, ?s中任一向量都可由 线性表示, , …, ? i1 ? , ? i2 ir 极大线性无关组(maximal linearly independent subset). 为?1, ?2, …, ?s的一个 , …,