TITLE: Why vectors in Linear Algebra start from point (0,0)?
QUESTION [9 upvotes]: 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 $(0,0)$?

REPLY [2 votes]: 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.

REPLY [2 votes]: 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 $(a,b)$.