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I was looking for what actually vectors are from linear algebra point of view. And how they differ from physics student perspective and computer science student perspective. I was watching a video 1 from 3Blue1Brown channel on linear algebra. And found that in linear algebra almost in all cases vector will be rooted at the origin.

Well, my question is why vectors are rooted at the origin in Linear algebra? As far as I know every convention in math has its purpose in reality. If it is a convention to be all vectors rooted at the origin, then what actually it means in human interpretation? Explanation with example is appreciated.

Similar question was asked before, here 2. But it didn't make me clear.

F.C. Akhi
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  • Take e.g. the vector $1$ of $\mathbb{R}$ over $\mathbb{R}$. How is it "rooted at the origin"? – NeitherNor Jul 26 '20 at 21:14
  • From linear algebra point of view, vectors can be added and multiplied by scalars, as long as they follow certain properties; they could be polynomials, functions, matrices, etc. – J. W. Tanner Jul 26 '20 at 21:19
  • @NeitherNor What actually you referring as vector 1 here? As far as I know ℝ is real coordinate space. And you didn't put any exponent over ℝ. So I presume its one dimensional or real number line. Am I correct? – F.C. Akhi Jul 26 '20 at 21:19
  • @J.W.Tanner Yes, I know those operations could be done over vector. But why all time rooted at the origin? Is it a property of vector in linear algebra? – F.C. Akhi Jul 26 '20 at 21:23
  • Exactly, I mean what you call a 1D vector. As (1,2,3) is a vector in $\mathbb{R}^3$, 1 is a vector in $\mathbb{R}$ (both times over $\mathbb{R}$). So how is 1 rooted at the origin, i.e. 0? – NeitherNor Jul 26 '20 at 21:34
  • @NeitherNor Hmm logically it should be zero – F.C. Akhi Jul 26 '20 at 21:40
  • @NeitherNor okay I got confused with my answer again – F.C. Akhi Jul 26 '20 at 21:46
  • No, this is not my question. My question is: in which sense is 1 rooted at 0? Your word "rooted" is not standard terminology, and I want to understand what you mean with it. If you mean that when you draw an arrow that it starts at the origin, then this is a physics convention, not a math one. – NeitherNor Jul 26 '20 at 21:46
  • @NeitherNor "If you mean that when you draw an arrow that it starts at the origin" Yes I exactly meant that. If starting at the origin is the physics convention, then in 3Blue1Brown linear algebra video 1 which I linked in my question says it's math convention not physics!!! – F.C. Akhi Jul 26 '20 at 21:51
  • Ok, let's try to be precise: in general, you cannot represent vectors as arrows. Thus, there is also no rule where these vectors should start. In $\mathbb{R}^2$, for example, you however can, and there is the convention, not the rule, to draw them from the origin if there is no other point which makes more sense. But there is no meaning attached to where the arrow starts. Now, in some parts of physics and engineering, there is a meaning attached to where these arrows start. Strictly speaking, these arrows there however don't represent vectors in the math sense... maybe pairs of vectors... – NeitherNor Jul 26 '20 at 22:00
  • @NeitherNor Thanks a lot. Now I got the point – F.C. Akhi Jul 26 '20 at 22:08
  • @NeitherNor I think „vectors not rooted in 0“ are called points in an affine space, at least if one considers the definition of affine space https://en.m.wikipedia.org/wiki/Affine_space – Jonas Linssen Jul 27 '20 at 07:04
  • @PrudiiArca I think it's best to think of vectors simply as some elements of an abelian group (which is how they are defined. There is some field involved, too, but it's unimportant for the discussion). There are no points or whatever, just this group. There is, by definition, an identity $e$ in this group such that $v+e=e+v=v$, but I wouldn't say that anything is rooted at $e$. You can, of course, start talking about affine spaces and who knows what, adding extra structure. But this extra structure will not change the definition of vectors. – NeitherNor Jul 27 '20 at 08:28

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