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I've got a good grasp on the definition and meaning of linear dependence and independence. If you have a set of vectors, and one of those vectors can be replicated via a linear combination of the other vectors in the set, you would call the call the vector set linearly dependent.

However, I'm somewhat curious on why it is called "dependent". What exactly is dependent? When I think of the word dependence, it makes me think that something is affected by the actions of something else (e.g. if y = f(x), y is dependent on x). However, I don't see what the dependency is in linear dependency. What exactly is the dependency?

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    $\begingroup$ The vector that you can get from the other ones is the dependent one. The choice of which one to make the dependent one is not unique. $\endgroup$
    – John Douma
    Dec 22, 2018 at 22:12
  • $\begingroup$ @JohnDouma That's good to the know that a vector is dependent and not the set. That's something new to me. However, my original question still stands, why is this vector considered "dependent" rather than just "unnecessary"? (I'm not advocating the term 'unnecessary', I'm just wondering why 'dependent' is the correct term) $\endgroup$
    – Izzo
    Dec 22, 2018 at 22:36
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    $\begingroup$ You are asking a linguistics question. My guess is that dependent is more accurate. I am not sure dependent vectors are unnecessary in every application. Would we then call independent vectors necessary vectors? $\endgroup$
    – John Douma
    Dec 22, 2018 at 22:37
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    $\begingroup$ I often introduce the concept in class in terms of redundancy and irredundancy. But it's fine to say that the one that's a linear combination of the others is dependent upon them. $\endgroup$
    – Ted Shifrin
    Dec 22, 2018 at 23:33
  • $\begingroup$ A set of vectors is linearly dependent iff some member of the set is composed (a linear combination) of—i,e., dependent on—all the other members. $\endgroup$
    – ryang
    Mar 27, 2021 at 12:18

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I'd think about it in terms of a dependence relation you can get. That is, let $\{v_1, v_2, \ldots, v_n\}$ be linearly dependent. Then there exist scalars $a_1, a_2, \ldots, a_n$, not all zero, so that

$$a_1v_1 + a_2v_2 + \ldots + a_nv_n = 0.$$

The above equation is a linear dependence relation, or a linear dependency. There are infinitely many choices for scalars $a_1, \dots, a_n$ satisfying this equation, but at the same time, not every set of $n$ numbers will work. You could say that the values for some subset of your $a_i$'s depend on your choices for the others.

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