Question

为什么 两两正交,非零的向量组必线性无关

Answer 1

设η1,η2,...,ηs是两两正交的非零向量.

k1,k2,...,ks是组合系数,使得k1·η1+k2·η2+...+ks·ηs = 0.

对i = 1,2,...,s,两边与ηi做内积得ki·(ηi,ηi) = 0 (向量两两正交,即对j ≠ i,有(ηj,ηi) = 0).

又ηi ≠ 0,故(ηi,ηi) ≠ 0,于是ki = 0,i = 1,2,...,s.

因此使η1,η2,...,ηs线性组合得0的系数全为0,即η1,η2,...,ηs线性无关.