In the wikipedia article about vector space I do not understand this sentence
Roughly, affine spaces are vector spaces whose origin is not specified.
A vector space does not need an origin. When one writes:
$\vec{v} = \pmatrix{v_x\\v_y\\v_z} = \pmatrix{1\\-4\\7}$
it only needs a basis, not an origin. Am I wrong, or what am I misunderstanding ?
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"The origin" is usually defined as a vector with coordinates $(0,0,\ldots,0)$ for some chosen basis.
It's not hard to see that these are the coordinates of the zero vector no matter which basis you choose for the vector space.
In ordinary vector algebra, "points" are identified with the tips of position vectors. A point is identified with the components of the vector, and vice versa.
Terminology and viewpoint change if you move from a vector space to an affine space, but the above explanation holds for vector spaces.
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Look at the definition of Affine space. It turn out that by choosing $a \in A$ you can identify the space with its underlining vector space. So basically you can choose what point turns out to be your origin.
Informally speaking, when you choose a point $a \in A$ (and fix it), you can identify vector and points in a natural way: for every point $b \in A$, $b-a$ is a vector of $V$ (vector space under $A$), and conversely, for every vector $v \in V$ there is a point $b \in A$ such that $a+v=b$ (look at the definition of affine space.
Informally speaking $x2$. An affine space is like a translated vector space, and its elements are points. To regains vectors from point, you need to translate it back to an origin
A vector space is a set whose elements are regarded as 'vectors', with addition and multiplication by scalars, satisfying the usual axioms.
A vector space always contains a zero vector which is the neutral element for addition ($0+x=x$ for all vectors $x$).
When we introduce vectors in the geometrical space (more generally, in an affine space), first we have to fix a point $O$ and call it 'origin', and then the vectors can be identified with the points of the space (by starting all vectors from $O$).