Question

一道线性代数题,Show that there is only one way to write v as a comb

Answer 1

设基向量组为e1,...,en,v有两种表示：

v=a1e1+a2e2+...+anen

且v=b1e1+b2e2+...+bnen

两式相减得

(a1-b1)e1+(a2-b2)e2+...+(an-bn)en=0

由于基向量组为e1,...,en线性无关,所以a1-b1=a2-b2=...=an-bn=0

即a1=b1,a2=b2,...,an=bn

that is,there is only one way to write v as a combination of the basis vectors