The coordinate of an event in a given system of coordinates is not the same thing as a vector between the origin of the coordinate system and the coordinate of the event; the coordinate itself is not a vector.  The difference between the coordinates of two infinitesimally separated events is a vector.  If the coordinates of both events are both transformed linearly to coordinates in a second coordinate system, each of the coordinates transforms with the same transformation, which may be a linear non-homogeneous transformation.  However, the additive constant that accounts for the non-homogeneity of the transformation law is the same for both of the coordinates.  So, the additive constants cancel each other when the difference between them is formed.  Thus, the linear non-homogeneous transformation of the coordinates becomes a linear homogeneous transformation of the coordinate differential, the vector.  This is a fundamental aspect of the theory of relativity. The coordinate differential is representative of all true vectors.  Vectors transform under linear homogeneous transformations even when the two coordinate systems are related by a linear non-homogeneous transformation.

Vectors are tensors of the first rank. A second rank tensor can be thought of as a vector whose components are vectors.  Higher rank tensors can be formed similarly.  Tensors of any rank can be expressed in different coordinate systems by applying the appropriate linear homogeneous transformation.  By describing natural events using field theories expressed with tensors, the field theories are invariant to linear transformations between coordinate systems.