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For a system of linear equations in $\Bbb R^n$ to be linearly independent, there must be a unique solution to the system (at least I'm pretty sure that's true). There are definitely other definitions, but this is the one I am most used to.

Nonetheless, I am confused! Why should a set of vectors be called linearly independent under these circumstances? I mean, what are they independent of? Ultimately, I would like to know why we use the terms linear dependence and independence?

Any help is appreciated. Thank you!

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    $\begingroup$ If one of the vectors linearly depended on the others, then you would have infinite solutions. Having a set of linearly independent vectors implies that a unique solution exists $\endgroup$ – jameselmore Sep 5 '15 at 5:06
  • $\begingroup$ in $\mathbb R^{2}$ linearly independent vectors are not along the same line . May be that has something to do with the name. Although dimensional spaces the linearly independent vectors are not lying all in the same hyper-plane. Just a thought. $\endgroup$ – user118494 Sep 5 '15 at 5:27
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    $\begingroup$ To me linearly independence is one of those math definitions that it is easier to get the feeling with characterisations (in this case something that isn't linearly dependent), rather than jumping into the definition itself. I find the definition of linearly dependence is intuitive enough. Often we need vectors that are NOT linearly dependent, and we call them linearly independent. There properties that characterise linearly independence, and many books use them as def. of linearly independence without using linearly dependence, which I find harder to get the feeling. $\endgroup$ – shall.i.am Sep 5 '15 at 6:47
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We have linear which is self-explanatory - 'of lines', and independence which means not reliant on each other, and the dictionary.com references the archaic definition competence.

So a linearly independent set of vectors is a set of lines that competently (necessary and sufficient) defines a vector space.

I mean, what are they independent of?

They are independent of each other.

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  • $\begingroup$ Which definition of "competence" is relevant here? Your second paragraph appears to confuse "linearly independent set" for "basis". $\endgroup$ – hmakholm left over Monica Aug 9 '19 at 9:17
  • $\begingroup$ @HenningMakholm; I read the question 'For a system of linear equations in $\mathbb{R^n}$ to be linearly independent' as implying we have $n$ (linearly independent) equations in $\mathbb{R^n}$. $\endgroup$ – JMP Aug 9 '19 at 9:20
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An expression of the form $a_1V_1 + a_2V_2 + ... + a_nV_n$ for vectors $V_i$ and scalars $a_i$ is called a "linear combination" because it generalizes the property that $y = ax$ is a line through the origin in the plane. It is simply a description that someone attached to the concept in the past, and is now common use.

A set of non-zero vectors are linearly dependent if at least one of them can be written as a linear combination of the others. It depends on the others. As such, you could kick that vector out as redundant, as everything that is a linear combination of the full set is also a linear combination of the other vectors without it.

A set of vectors is linearly independent if none of its vectors is a linear combination of the others. In this set, there are no redundant vectors. Throw out one, and you get a smaller span.

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  • $\begingroup$ I have come to the conclusion that "linearly redundant set" would be a better, more descriptive label for what is called a "linearly dependent set." $\endgroup$ – Selrach Dunbar Feb 19 at 18:05
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The elements are dependent on each other if one term can be expressed in terms of the other terms. Otherwise, they are independent of each other.

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