let be a vector space such that and let be bases of V, the transformation matrix from to is defined as where each represents a column vector
let us take a manifold with dimension . we will denote the components of a vector with the numbers . if one modifies the basis, in which the components of vector are expressed, then these components will cchange, too. such a transformation can be written using a matrix , of which the columns can be regarded as the old basis vectors expressed in the new basis ,
note that the first index of denotes the row and the second index the column. we call a transformation matrix.
consider the following bases of :
find the following:
instead of putting each of the first 3 in its own matrix and reducing we use 1 matrix for them all
which gives us:
let be a vector space such that and let be bases of then:
finding transformation matrix of a linear function
if one has a linear transformation in functional form, it is easy to determine the transformation matrix by transforming each of the vectors of the standard basis of the domain of by , then inserting the result into the columns of a matrix, so we get: