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Could you give some examples? I know that let's say $\{[1,0], [0,1]\}$ is a maximal independent set of vectors in $\mathbb R^2$ because no vector can be added to the set from $\mathbb R^2$ that would be linearly independent with the basis. But what is the maximum independent set in this case?

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    $\begingroup$ There are topics in which these two words are used differently (very confusing). But in this case I think they mean the same thing. $\endgroup$ – David Dec 4 '15 at 3:46
  • $\begingroup$ Yes, I think in the same way. However, is there any example in this context where the maximal set would be different from maximum? $\endgroup$ – ady Dec 4 '15 at 3:49
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    $\begingroup$ @David : I think the standard way in which these two words are used differently from each other fits this situation very well. See my answer below. ${}\qquad{}$ $\endgroup$ – Michael Hardy Dec 4 '15 at 3:50
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There isn't any.

  • A "maximum" linearly independent set would be a linearly independent set that includes every other linearly independent set as a subset. That can't happen because you can find another linearly independent set that is not a subset of this one, and of which this one is not a subset, and you can't find any third linearly independent set of which they are both subsets, since that would contain more than two vectors in a two-dimensional space.
  • A "maximal" linearly independent set just means a linearly independent set that is not a subset of any other linearly independent set.
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  • $\begingroup$ I was thinking that every maximal independent set would also be maximum in this case. So, maximum in terms of size = 2. Moreover, I thought that if there exists a maximal set then there also exists a maximum set. $\endgroup$ – ady Dec 4 '15 at 4:05
  • $\begingroup$ of course maximum $\rightarrow$ maximal $\endgroup$ – ady Dec 4 '15 at 4:08
  • $\begingroup$ In the graph theory, we have: a graph in which every maximal independent set has the same size is a graph in which every maximal independent set is maximum en.wikipedia.org/wiki/Well-covered_graph $\endgroup$ – ady Dec 4 '15 at 4:12
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Every set of linearly independent vectors is maximal and maximum. Thus, in the example above, the set of linearly independent vectors {[1,0],[0,1]} is maximal and maximum.

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  • $\begingroup$ It's not "maximum" because it is not the case that every linearly independent set is a subset of this one. It is, however, maximal, since you can't add any more vectors to it and still have it be linearly independent. $\endgroup$ – Michael Hardy Dec 4 '15 at 19:30
  • $\begingroup$ I think it depends on the interpretation. If we assume that the maximum is the largest independent set in size in 2D, then it is correct. $\endgroup$ – ady Dec 5 '15 at 5:59
  • $\begingroup$ It's not the largest one in the ordering by inclusion, i.e. it is not the case that all other independent sets are included in it. That is a standard usage when speaking of partially ordered sets. ${}\qquad{}$ $\endgroup$ – Michael Hardy Dec 5 '15 at 16:42

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