>>14360608OP, a linear transformation is a function that maps vectors in to vectors in with the property
consider the basis expansion of a vector
where the basis vectors have subscripts and the coordinates have superscripts (they are not being raised to the i-th power)
expand the vectors in the linear map in terms of some bases
and by the property of linearity we see
this doesn't seem to get us anywhere, but note you can expand the transformed basis vectors
rearranging this expression gets you
and finally invoking the uniqueness of basis representations we find
the coefficients form the matrix.
it's easy to see that composing two linear transforms nets you matrix multiplication
where
it's all linear transforms and basis expansions.
