Question

线性代数，Ax=0求解，请问有解吗？有解求过程。如果无解，请问为什么？

A
1-1 0 0 1
0 0 1 1 1
1 1 1 0-1
1 0 0 1 0

Answer 1

A =
[1 -1 0 0 1]
[0 0 1 1 1]
[1 1 1 0 -1]
[1 0 0 1 0]
初等行变换为
[1 -1 0 0 1]
[0 0 1 1 1]
[0 2 1 0 -2]
[0 1 0 1 -1]
初等行变换为
[1 0 0 1 0]
[0 1 0 1 -1]
[0 0 1 1 1]
[0 0 1 -2 0]

初等行变换为
[1 0 0 1 0]
[0 1 0 1 -1]
[0 0 1 1 1]
[0 0 0 -3 -1]
初等行变换为
[1 0 0 1 0]
[0 1 0 4 0]
[0 0 1 -2 0]
[0 0 0 3 1]
取 x4 为自由未知量，方程组化为
x1 = -x4
x2 = -4x4
x3 = 2x4
x5 = -3x4
取 x4 = -1， 得基础解系 (1, 4, -2, -1, 3)^T,
方程组通解是 x = k (1, 4, -2, -1, 3)^T。