例2证明n维向量组 E1=(1,0,…,0),2=(0,1,…,0),…,En=(0,0,…,1) 线性无关 证明设k11+k22+…+knn=0,由于 k1 81+k62 k, 8 =k1(1,0,…,0)+k2(0,1,…O)+…+kn(00,…,1) (k1,k2,…,kn)=0, 故k=k2=…=kn=0,所以1,62…,En线性无关例2 证明 n 维向量组 ε1=(1,0, ···,0), ε2=(0,1, ···,0), ···, εn=(0,0, ···,1) 线性无关. 证明 设 k1 ε1+k2 ε2+ ···+kn εn =0, 由于 k1 ε1 + k2 ε2 + ···+ kn εn = k1 (1,0, ···,0)+k2 (0,1, ···,0)+ ···+kn (0,0, ···,1) = (k1 , k2 , ···, kn ) = 0, 故 k1 = k2 = ···= kn = 0, 所以 ε1 , ε2 , ···, εn 线性无关