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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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4 Answers

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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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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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A different style of answer -

Imagine a space, in which lives our family of, say, four vectors. Vector A and B or very independent, but C and D depend on each other - you could take C, and get D from him, or take D, and get C from him. More formally, one can be expressed from the other.

Imagine the space being ripped apart, and a part of it being taken away. If the ripping apart took away A and the space that supported him, then B can keep on existing. But if it took away the space that C lived in, D cannot live in this space anymore - a fundamental part of him was taken away.

So, in a way, D depends on the existence on C in the space. If the space cannot contain C, it cannot contain D, and vice versa. But A and B still remain expression-able in the space even if C and D do not.

this style of answer thinks of the space foremost and the list of vectors last, which is contrary to the usual mode of thought. But both are one and the same, depending on your current perception. so take whichever intuition helps you the most.

Cheers, and good luck!

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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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