Problem about equivalent conditions for a basis of a free module

In summary, a free module is a module that can be generated by a linearly independent set of elements. Equivalent conditions for a basis of a free module include linear independence, uniqueness of linear combinations, and the ability to express any element in the module as a finite linear combination of basis elements. To prove a set of elements is a basis for a free module, it must be both linearly independent and able to span the entire module. A free module can have more than one basis due to the non-uniqueness of bases. Understanding equivalent conditions for a basis of a free module is important for determining if a set of elements can serve as a basis and identifying different bases for the same module.
  • #1
Ganitadnya
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Here's the problem:
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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.
 

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  • #2
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)).
 

1. What is a free module?

A free module is a module that has a basis, which means it can be generated by a linearly independent set of elements.

2. What are equivalent conditions for a basis of a free module?

Equivalent conditions for a basis of a free module include the following: 1) every element can be uniquely written as a linear combination of basis elements, 2) the basis elements are linearly independent, and 3) every element in the module can be expressed as a finite linear combination of basis elements.

3. How do you prove that a set of elements is a basis for a free module?

To prove that a set of elements is a basis for a free module, you need to show that the elements are linearly independent and span the entire module. This can be done by showing that every element in the module can be written as a unique linear combination of the basis elements.

4. Can a free module have more than one basis?

Yes, a free module can have more than one basis. This is because a basis is not unique and there can be multiple sets of linearly independent elements that can generate the same module.

5. What is the importance of understanding equivalent conditions for a basis of a free module?

Understanding equivalent conditions for a basis of a free module is important because it helps us determine whether a set of elements can serve as a basis for the module. It also allows us to identify different bases for the same module and understand the relationship between them.

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