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I am studying linear independence in introductory linear algebra, and a doubt crossed my mind, which I wanted to clear. Suppose, we have a set on N vectors, all of same dimension. (N is a constant)

Say, we want to reduce this set of N vectors into a subset of N1 independent vectors. Then the remaining (N-N1) vectors would each be dependent on these N1 vectors, and can be expressed as their linear combination.

Now, it is not necessary that the original set of N vectors can be reduced to just one subset of independent vectors. We can have many such subsets.

My question : Is the number of independent vectors in all these subsets equal? i.e. do they all have same value of N1?

Kindly note that it would be much appreciated if someone could give an elementary explanation without using the concept of rank, or vector spaces, as in the textbook I'm studying, those topics are all covered after linear independence.

Thank you!

Sumit
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    The short answer is Yes. The longer answer will need some discussion of vector spaces, as the key result is that the dimension of a vector space is well defined. In particular, your independent collections each form a basis of the span of your original set, so the size of each such collection is the dimension of that span. – lulu Feb 08 '20 at 13:28
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    See, e.g., this question for a discussion of the basis result. – lulu Feb 08 '20 at 13:32
  • Thanks @lulu! That clarifies it. – Sumit Feb 08 '20 at 13:32

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