Frames of references and coordinate systems In linear algebra, a vector can be represented by different bases. However, this is merely a different representation of the same entity; i.e. $\vec x = x\hat\imath + y\hat\jmath + z\hat k = x'\hat\imath' + y'\hat\jmath' + z'\hat k'$. The basis vectors can be related by $\hat\imath = \hat\imath'R$ etc... (where $R$ is a transformation matrix), which means both sets of basis vectors belong to the same vector space.
In a given frame of reference, a point is specified by a position vector, $\vec x$, relative to the origin of the frame, with respect to a choice of basis vectors - $\{\hat\imath,\hat\jmath,\hat k\}$. If the same point is specified in another frame by position vector $\vec x'$, with an origin displaced by a vector $\vec r$ relative to the original, which uses the same basis vectors - $\{\hat\imath,\hat\jmath,\hat k\}$, then the vector $\vec x \ne \vec x'$, but rather $\vec x=\vec x'+\vec r$. Does this mean that reference frames must agree on a vector space, and/or is that an observer in a given frame chooses a vector space in that frame, and then imposes this on all other frames? For example, if an observer in a given frame chooses the $(1, 0, 0)$ to point in particular direction, would this now be consistent across all frames?
 A: A vector space is intrinsically defined and requires no choice of a frame to specify a vector.
However, it is common to define a vector via first choosing a frame and then specifying a coordinate vector. These two pieces of information then specifies a vector.
If we then choose another frame, then this same vector will have a different coordinate vector but the two will be related. In fact, the relation is as follows:

$v^f = T^f_e . v^e$

Here, $v$ is the choice of vector. And $e,f$ are the names of the two frames. And $v^e, v^f$ are the coordinate vectors of $v$ in the frames $e,f$ respectively. Finally, $T^f_e$ is the change of frame matrix, changing frames from $e$ to $f$.
Now, your change of frame includes a translation. And this suggests that you are thinking of affine spaces rather than vector spaces. These are a generalisation of vector spaces that do not require an origin. Vector spaces always come with an origin, the zero vector. Affine spaces are more natural in physics. However, vector spaces are more natural in mathematics - they have a simple axiomatic definition. And generally, affine spaces are commonly understood via an action by a vector space: the vector space of displacements.
A: The typical approach is to break the problem up into a frame, which has an associates vector space, and a coordinate system which defines a basis.
There is no general way to equate vectors in one frame with vectors in another.  This is most obvious one frame that rotates with respect to another.  If we look at a velocity vector, we find that the transformation to the other space requires a position vector to get the centrifugal terms right.  So we cannot even say there is a meaningful bijection between the velocities - that mapping is position dependent.
In the simple case you describe, where one frame is a simple translation from the other, these mappings can be made, and are reasonably straight forward (other than position maps differently than higher derivatives do).  But in general the vector spaces associated with different frames are quite unrelated.
