affine transformation matrix
consider the following simple equation:
we can define the (infinite) set of vectors that these equations span
which can also be written in the usual function form
note that this isnt a linear transformation, but an affine transformation, because 
a transformation matrix can only be deduced for linear maps, we can turn 
into a linear map if we drop the intercept which is keeping it from preserving the origin:
now that we know for sure 
is a proper linear function we can treat it as such
the standard basis of the domain of is 
we apply to these vectors:
we place these in a matrix (as column vectors) to get the transformation matrix for
now we bring back the intercept, to transform a given vector using the matrix formula would be:
for convenience, we use homogeneous coordinates to turn this formula into just one matrix multiplication by adding one column and one row and placing the intercept in the last column with an additional 
to denote that its a point (homogenous coordinates notation):
we can define the (infinite) set of vectors that these equations span
which can also be written in the usual function form
note that this isnt a linear transformation, but an affine transformation, because
a transformation matrix can only be deduced for linear maps, we can turn into a linear map if we drop the intercept which is keeping it from preserving the origin:
now that we know for sure is a proper linear function we can treat it as such
the standard basis of the domain of is we apply to these vectors:
we place these in a matrix (as column vectors) to get the transformation matrix for
now we bring back the intercept, to transform a given vector using the matrix formula would be:
for convenience, we use homogeneous coordinates to turn this formula into just one matrix multiplication by adding one column and one row and placing the intercept in the last column with an additional to denote that its a point (homogenous coordinates notation):