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!