I have been learning about matrices, and am interested in their use as maps. I have seen that there is in fact a matrix for transforming to spherical and cylindrical polar coordinates (Wikipedia), whose determinant gives the scaling factor of the volume element (which is slightly confusing to me since I thought this was the role of the Jacobian, and the transformation matrix doesn't look like the Jacobian. Anyway...)
I was wondering if there are maps that matricees cannot represnet. I am sure there are in abstract mathematics, but I mean mappings that would be useful in physical problems such as that from cartesian coordinates to spherical or cylindrical polars, or some other curvilinear coordinate. Has this to do with the linearity of matrix transformations? From what I understand, matrices all represent linear maps because they follow the two rules of linear transformations.
EDIT: I have now seen this post in which it is said that non-linear transformations cannot be respresented by matrices, except by going to an n+1 dimensional matrix. So I suppose that answers part of my question. The things that are still not clear to me are:
- Is it the case that EVERY matrix represnets a linear transformation?
- and, can EVERY linear transformation be respresented by a matrix?
- What restrictions are there as to which non-linear transformations can be represented by matrices (or indeed, which non-linear transformations can be represented by matrices if not all can).