Problem about equivalent conditions for a basis of a free module

[Ganitadnya]

Here's the problem:
View attachment 7922

I don't see why there should be only finitely many nonzero a_z in b. I was able to prove uniqueness assuming that there only finitely many nonzero. I was able to show b implies d and b implies c, c implies a.

 

[Euge]

So you're referring to the implication (a) $\implies$ (b). There are only finitely many $a_z$ in (b) simply because $Z$ is a generating set for $M$ (since it is a basis by (a)).