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I originally learned vectors as something with a magnitude and direction, which meant that a vector could be translated, moved along a coordinate system.

But, they are actually just an n-set of coordinates. I recently realized, doesn't a set of coordinates give more information than just a magnitude and direction? They also give information relative to the origin, so vectors shouldn't be able to translate through a coordinate system, right?

Also, I'm familiar with representing things like forces with vectors as something with a magnitude and direction. I don't conceptually understand how to use a set to represent forces and other vector quantities.

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  • $\begingroup$ The elements of a vector space are vectors. For example functions can be viewed as vectors. They are also used as those things that aren't scalars. I think you are referring to coordinates are tied to a particular basis of the vector space. $\endgroup$ – Karl Oct 29 '17 at 7:12
  • $\begingroup$ Coordinates could be considered to give information about the head of the vector relative to its tail. Then the coordinates of a vector wouldn't change if you moved the whole thing (head and tail) around as long as you didn't stretch or turn it. $\endgroup$ – Joshua Meyers Oct 29 '17 at 7:13
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    $\begingroup$ Great question! In one way of looking at this, there’s actually a separate vector space at each point, but for a simple Euclidean space, they’re all identical and you can jump from one space to another—“move” the vectors—without worrying about this fine point. You can also largely get away with blurring the difference between points and vectors. However, when you start studying more complex structures, you’ll find that these tangent spaces can be different from point to point and you’ll have to be much more careful about these things. $\endgroup$ – amd Oct 29 '17 at 8:05
  • $\begingroup$ Also, to reinforce the point made by @Karl, vectors aren’t “just an $n$-set of coordinates.” For example, linear functions that take a vector and spit out a number are perfectly good elements of a vector space. However, once you choose a basis for the vector space—i.e., pick a coordinate system—then you can identify vectors with coordinate tuples. $\endgroup$ – amd Oct 29 '17 at 8:13
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from what you said i assume that you take vector $\vec v\in\Bbb R^n$ to be $\vec v=\begin{bmatrix}v_1\\v_2\\\cdot\\\cdot\\\cdot\\v_n\end{bmatrix}$

this way is a way to show vectors that the starting point of them is the origin. BUT you have to keep in mind that this is just a way for US to look at it, the origin of the axis is not a set point, it is a point we decide upon.

let's say i want to move the vector in the first axis(change only $v_1$) i can say that i have a vector $\begin{bmatrix}1\\0\\\cdot\\\cdot\\\cdot\\0\end{bmatrix}+\begin{bmatrix}v_1\\v_2\\\cdot\\\cdot\\\cdot\\v_n\end{bmatrix}$ now this vector is not the same as $\vec v$ if you really add those $2$ but if you look at the first vector as a way to move the origin, you have the vector $\vec v$ with the starting point of the origin$+\begin{bmatrix}1\\0\\\cdot\\\cdot\\\cdot\\0\end{bmatrix}$. so to summarize, this way to represent vectors is just a convenient way to show all the information that a vector has

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