向量α1,…,αk线性相关，证明对α，有λ1,…,λk，使α1+λ1α,…,αk+λkα线性相关

wilsony

谁能证明？

elim

设\(\,\eta_j\)不全为零使得\(\,\eta_1\alpha_1+\cdots+\eta_k\alpha_k=0.\) 考虑\(\lambda_j\)的取值.
若\(\eta_j\)全不为\(0\), 取\(\lambda_1=\eta_2,\,\lambda_2=-\eta_1,\,\lambda_j=0\,(j>2);\)
若有某\(m\in\{1,\ldots,k\}\)使\(\eta_m=0\), 取\(\,\lambda_m=1,\;\lambda_j=0\,(\,j\ne m)\)
故总有\(\lambda_1,\ldots,\lambda_k\)不全为\(0\)且\(\displaystyle\sum_{j=1}^k\eta_j(\alpha_j+\lambda_j\alpha_{k+1})=\big(\sum_{j=1}^k(\eta_j\lambda_j)\big)\alpha_{k+1}=0\)

luyuanhong

wilsony

多谢陆大师和elim大师！