矩阵的秩4(双语教学).doc

文档介绍：Act 3  of  Chapter 3
【Content And Arrangement】
1) Rank of matrix
2) Vector space(一)
【Content 】
Section 4   Rank of matrix (矩阵的秩)
I.  Definition
1. Minor
   在矩阵A中任取k行,k列(),位于这k行、k列交叉位置上的元素按原来的次序构成的k阶行列式,称为A的一个k阶子式.
注: k 阶子式的个数是
2. Rank of matrix:
The largest order in the whole nonzero minor of matrix A is called rank of A,namely R(A) .
Example8: If,then R(A)=?
Solve: 。它的全部3阶子式(共有个3阶子式)
      为
左上角的2阶子式, 故.
?
 Theorem1:R(A)=R(AT).
 Theorem2:初等变换不改变矩阵的秩.
  (1)上阶梯形矩阵:形如
 并且上阶梯形矩阵应满足的条件:
(a)若有零行,都应在矩阵的下方;
(b)每一行的第一个非零元的下方均为0.
如:
                     
(2)上阶梯形矩阵的秩:非零行的行数.
(3)利用初等行变换求矩阵的秩:
上阶梯形矩阵B, 则.
  Example9:求矩阵A的秩:  
  Solve: 对 A 进行初等变换:   
            
   则 R(A) =3.
II: The relation between rank of vector bundle and rank of matrix
Row vector bundle and column vector bundle:
  If
     then is called a row vector bundle of A,    and  is called a column vector bundle of A.
2.  Row rank : Rank of a row vector bundle of A is called row rank of A .
Column rank:Rank of a column vector bundle of A is called column rank of A.
Results: R(A)= row rank of A =column rank of  A .
?
   设向量组,将以行或者列构成一个矩阵 A ,则由上述结论可知: 的秩= R(A).
Example10:Judge the following vector bundle's linear dependence or linear independence. Give its maximal linear independent bundle and its rank.
Solve: 设
  通过计算, R(A) =3 <4,则向量组的秩为3,且线性相关.
在 A