Linear-Algebra
Space has no intrinsic idea of a grid/coordinate system this means that the basis Vectors are just convention but you could describe the coordinate system - and in turn Matrices/Linear Transforms etc.. - with other basis vectors. In general in dimensions you can pick any vectors and describe the whole space - as long as they are Linearly Independent.
Given a non-standard set of basis vectors we can find how to write any vector in terms of our standard bases . By taking a vector expressed in terms of we find in terms of by applying the linear transform of matrix where the column vectors of are the basis vectors .
We can find how to express a vector in terms of by multiplying by .
To express a transform (that is in terms of ) in terms of we simply compute the matrix
the logic behind this is that we first change our vector into a vector in terms of then apply the matrix (which is in terms of so they are compatible) then we turn the newly transformed vector back into our coordinate system by multiplying by .