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I learn linear algebra in university and I was wondering why vectors in linear algebra always start from the point $(0,0)$?

how many kinds of mathematical vectors out there?

Is it legit to use other kind of vectors in linear algebra apart from that vectors that start at $(0,0)$?

Emilio Novati
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LiziPizi
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    There are so-called displacement vectors associated to so-called affine spaces which begin at one point in an affine space (for example, at one point in a plane in $\mathbb{R}^3$) and end at another. A common example that gets drawn in calculus is the normal vector to a plane not through the origin, which gets drawn starting at a point in the plane. However, you can always translate your coordinates to make the starting point into the origin if you want. – Ian Jun 03 '16 at 20:59
  • What do you mean by "always start from the point $(0,0)$"? I don't think of a vector as having a "starting point". When I think about a vector, I might visualize an arrow, but I keep in mind that any other arrow with the same magnitude and direction represents the same vector. – littleO Jun 03 '16 at 21:10
  • Well, if you consider the vector space $\mathbb{R}_{>0}^2$ as an affine space over itself, then the origin is $(1,1)$. – Björn Friedrich Jun 03 '16 at 21:27
  • @littleo I learned that in linear algebra in contrast to physics our vectors start at the origin and finish at some point, so a vector does not have only magnitude and direction but also a location (that is the end of the vector). I learned linear algebra only one year so I am not so experienced. – LiziPizi Jun 03 '16 at 22:42
  • "I learned that in linear algebra in contrast to physics our vectors start at the origin and finish at some point, so a vector does not have only magnitude and direction but also a location" I think this is simply wrong. – littleO Jun 03 '16 at 22:44
  • @littleo if I will call it length and not magnitude would it still be wrong? – LiziPizi Jun 03 '16 at 22:47
  • I think so. Vectors don't have a location. – littleO Jun 03 '16 at 22:49

2 Answers2

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Points are not elements of a linear space. If you want speak about points you have to work in an affine space, and, in this context, vectors can have origin at any point.

Emilio Novati
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  • I never said that points are elements of a linear space. btw, what is affine space? and does linear space only related to linear algebra? – LiziPizi Jun 03 '16 at 22:46
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    Intuitively, an affine space is a set of elements, called points, such that at any point is associated a vector space. The vectors of such vector space are defined as the ''difference'' between two points, so that, givena point P, for any other point $Q$ we have a vector $\overrightarrow{PQ}$ that, added to $P$ gives $Q$. So any point $P$ can be chosen as the origin of a vector space and the vectors of such vector space can be thinked as ''displacements'' (translations) from this origin point. – Emilio Novati Jun 04 '16 at 12:54
  • There a re different axiomatization of the affine spaces, you can see : https://en.wikipedia.org/wiki/Affine_space#Definition. – Emilio Novati Jun 04 '16 at 12:54
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Abstractly, vectors are not actually "arrows". The real way to define a vector is to define a vector space.

Let $V$ be a set, and $F$ a field. If we define a way to "add" the elements of $V$, and a way to "multiply" an element of $V$ by an element of $F$ that satisfy some basic nice properties like distributivity, then $V$ is called a vector space over $F$. The elements of $V$ are called vectors.

As an example let $V = \{(a,b) | a,b \in \mathbb{R}\}$, and $F = \mathbb{R} = \textrm{ the real numbers}.$

Define addition by $(a,b) + (c,d) = (a + c, b + d)$ and the scalar multiplication by $r(a,b) = (ra,rb)$ for $r \in \mathbb{R}$. Then $V$ is a vector space. In fact it is one that you are familiar with. The vector $(a,b)$ you can think of as the "arrow" from the origin to the point $(a,b)$.

Ken Duna
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