Question

线性代数相关证明 谢谢

Suppose that, for a given matrix A, there is a nonzero vector x such that Ax=0. Show that there is also a nonzero vector y such that A*y=0

Answer 1

证明: 由已知存在非零向量 x 满足 AX=0
取 y = 2x, 则 y ≠ x,
且 Ay = A(2x) = 2Ax = 0
即找到了另一个非零向量y满足 Ay = 0#

上面把A*y=0 以为是Ay=0了.
若A*是伴随矩阵, 就应该这样证明:

证: 由已知存在非零向量 x 满足 Ax=0, 所以齐次线性方程组 AX=0 有非零解.
所以 |A| = 0. (这是AX=0 有非零解的充分必要条件)
所以 |A*| = |A|^(n-1) = 0 (这是个知识点)
所以 A*X = 0 有非零解.
所以存在非零向量y满足 A*y = 0.

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说实话 我看不太懂中文术语 题就是这么个题 是我们作业 刚学。。。

追答

Proof: For there is a nonzero vector x such that Ax=0, So the equation AX=0 has nonzero solution.
So the rank of A is less than n.
So the rank of A* (Adjoint matrix of A) is less than n.
So the equation A*X=0 has a nonzero solution y.
then y is the nonzero vector satisfy A*y = 0.

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行 就凭这热心劲 得给你分。。

Answer 2

因为Ax=0有非零解
所以A一定不是满秩的
所以A*一定不是满秩的
所以A*y=0有非零解

Answer 3

Proof:
Lemma 1: Given a matrix M, a nonzero vector V, |M|=0 is a sufficient and necessary condition of MV=0.
Since AX=0(X≠0), we can know |A| = 0, therefore |A*| = |A|^(n-1) = 0, which implies we can choose a nonzero vector y such that A*y = 0.