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系判定

1.　讨论下列向量组的线性相关性

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3.用基础解系表示下列方程组的全部解.

(x1 , x2 , x3 , x4)T =c11 + c22 (c1, c2R).

x = c11 + c22 + ,

4.已知矩阵

3个向量构成,因此还需补充一个解向量,这个解

5 .k取何值时, 下列方程组无解? 有唯一解?

R(A) = R(B),

R(A) = R(B),

6.已知 Ax = b的三个特解为

(1)求对应的齐次线性方程组 Ax = 0 的通解;

(2) 求 Ax = b的通解;

(3) 求满足上述要求的一个非齐次线性方程组.

(1)由已知知方程组 Ax = b是含有

1 , 2为方程组 Ax = 0 的解,且 1 , 2线性无

(2)方程组 Ax = b的通解为

(3)因为

Ax = b的保留方程组只有1个方程, 设为

d为任意常数.

7.已知三维向量组:

(1) 可由 1 , 2 , 3线性表示, 且表达式是唯

(2) 可由1 , 2 , 3线性表示,但表达式不唯一.

(3) 不能由 1 , 2 , 3线性表示.

k1 , k2 , k3的线性方程组

(1)t取何值时, 方程组有唯一解;

(2)t取何值时, 方程组有无穷多解;

(3)t取何值时, 方程组无解.

(1)当

(2)当

(3)当

8.设 A为 mn矩阵, B为 n s矩阵, 若

AB = O, 试证: R(A) + R(B) ≤n.

R(B) ≤n-R(A),

9.则本题的三个问题可转化为以下的三个等价问题设 A，B均是 m n矩阵, 证明:

R(A + B) ≤R(A) + R(B) .

A有r个线性无关的行向量, 设为 1 , 2 , ··· , r ;

B有s个线性无关的行向量, 设为 1 , 2 , ··· , s ;

C : 1 , 2 , ··· , r , 1 , 2 , ··· , s

10.则本题的三个问题可转化为以下的三个等价问题设 n阶方阵 A满足 A2 = A, 试证:

R(A) + R(A-E) = n .

因为 A(A - E) = A2-A = O,

R(A) + R(A - E) ≤n.

= R(A) + R(E - A) ≥R(A+ E - A)

= R(E) = n .

11.则本题的三个问题可转化为以下的三个等价问题设 A是 m k矩阵, B是 k n矩阵, 试证:

R(AB) ≤ min { R(A) , R(B) } .

ABx = 0 的解空间的子空间.

n-R(AB) , 故

n则本题的三个问题可转化为以下的三个等价问题-R(AB) ≥n-R(B) ,

= R(BTAT) ≤R(AT) = R(A),

(则本题的三个问题可转化为以下的三个等价问题每小题8分，共24分)．

(则本题的三个问题可转化为以下的三个等价问题每小题8分，共24分)．

(12分)