Question

线性相关证明题If η is a solution of AX=b ,which and R(A)=r.Letζ1ζ2ζ

Answer 1

Suppose η,ζ1,ζ2,ζ3,ζ4,ζ5 are linearly dependent.

Then there are k1,k2,...,k6 of which not all are 0 such that k1η+k2ζ1+k3ζ2+k4ζ3+k5ζ4+k6ζ5=0.

So A(k1η+k2ζ1+k3ζ2+k4ζ3+k5ζ4+k6ζ5)=0.

Because Aη=b,Aζi=0.

So k1b=0,namely k1=0.

So k2ζ1+k3ζ2+k4ζ3+k5ζ4+k6ζ5=0.

Because ζ1,ζ2,ζ3,ζ4,ζ5 are a basis of the solution space of AX=0,

so they are linearly independent.

So k2=k3=...=k6=0

That is impossible.

So η,ζ1,ζ2,ζ3,ζ4,ζ5 are linearly independent