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So i have two questions:

1) If Vectors (say in R2) are independent of their location in space, then couldn't you theoretically for any vector in R2 regardless of its position, just move it such that its tail emanates from the origin to become a position vector?

2) Can we represent all positions/points in the 2D coordinate system as position vectors, and does that mean all points in the 2D coordinate system are just vectors emanating from the origin?

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    $\begingroup$ (1) Yep. (2) Yep. Not necessarily -- being able to represent $A$ by $B$ doesn't actually make $A=B$ (I can represent the number of whole dollars in my bank account by little pieces of scrap paper, but just because I collect thousands of pieces of paper doesn't mean I have thousands of dollars). $\endgroup$ – Bobbie D Feb 20 '17 at 4:10
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    $\begingroup$ 1) Yes. If you're a visual person, you can imagine things like linear dependence this way. (2) Yeah...but not always..and if you can, you probably shouldn't. For example, you could define a function $f: \mathbb{R}^2 \mapsto\mathbb{R}^2 $ (tail$\mapsto$head), to represent a set of cartesian coordinates(representing a square for example) you have in mind. But if you try to solve geometrical problems in this setting you are going to have an unnecessarily hard time. $\endgroup$ – Akay Feb 20 '17 at 6:08

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