Difference between changing coordinates and changing basis? If we have a vector $v$ then is has some coordinates w.r.t to a basis $B$ say $(a,b)$. What if we transform this into "polar coordinates"? Does this change the basis or just transform the coordinates into some new pair? Giving us new coordinates for the same vector in the same basis.
If we change basis I know we change coordinates, but can we change the coordinates and keep the basis? I am having troubles finding a basis for say polar coordinates which makes me suspect that this is just some kind of bijection and we still have the same vector.
 A: Changing coordinates linearly allows you to think of this as a change of basis on the whole Euclidean space. If you do some non-linear change of coordinates, like to polar, there is no such way to think about the coordinates themselves.
However, there is a notion of 'tangent space' to a point, which is a vector space of tangent directions. For curves, for example, the tangent space is a line. In Euclidean space, the tangent space at every point is just a copy of Euclidean space with the origin shifted to that point. Changing coordinates linearly induces the same linear change of coordinates on tangent spaces. However, for non-linear change of coordinates, one can compute the Jacobian matrix, which keeps track of the change of coordinates on the tangent space. If you evalaute the Jacobian matrix at the point, this gives the change of coordinates on tangent spaces.
If you think of the derivative (and hence the Jacobian matrix) as the best linear approximation, then you can think of this matrix as telling you the closest possible linear coordinates near the chosen point. This is, I believe, as good as you can get.
