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Answer by Lereau for Why doesn't the definition of dependence require that...

Let me try to give you some intuition on the term independence, since you are lacking one so far. The usual definition is indeed\begin{align}\{ v_1 , \dots, v_n \} \text{ are linearly independent}...

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Answer by baris for Why doesn't the definition of dependence require that one...

I think if you look at "dependent" as the negation of "independent" instead of the other way around, it'll make sense to you.Independent is the lack of any dependence. So if there is even the tiniest...

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Answer by user14972 for Why doesn't the definition of dependence require that...

Why is my intuition incorrect?I posit your intuition is incorrect because you learned from a biased source.You probably learned about the idea of independence from talk of (in)dependent variables...

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Answer by Ted for Why doesn't the definition of dependence require that one...

Your intuition for linear (in)dependence is very close. Based on your intuition, the definition you're looking for is:$\{v_1, ..., v_k\}$ is linearly dependent if there exists an index $i$ and scalars...

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Answer by zipirovich for Why doesn't the definition of dependence require...

Let me address your last question (and hopefully it will help with clarifying some of your misconceptions):Are they essentially the same definition except for this weird edge case?No, not only in that...

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Answer by Theo Bendit for Why doesn't the definition of dependence require...

I find that many of my students think the same way. Instead of thinking about null linear combinations, they usually prefer to think in terms of vectors as linear combinations of other vectors. And...

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Answer by Michael Hardy for Why doesn't the definition of dependence require...

To say that one of the vectors is a linear combination of the others singles out a vector to play a different role from the others. And it's possible that there are some among them that are not linear...

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Answer by Alex Ortiz for Why doesn't the definition of dependence require...

I would prefer you state your definition of linear independence thusly:Definition: The subset $\{v_1,\dots,v_n\}\subset V$ is linearly independent if whenever $a_1,\dots,a_n\in F$...

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Why doesn't the definition of dependence require that one can expresses each...

I was reviewing my foundations on linear algebra and realized that I am confused about independence and dependence. I understand that by definition independence means:A set of vectors...

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