Let the linear transformation from 2-d to 3-d space T defined by T(x,y) = (3x+y,x+y,x+3y).
Determine the matrix representing T relative to the standard basis for the codomain and the basis {(3,1),(4,1)} for the domain.
I am unsure of how to obtain such a matrix. My attempt consists of applying T on the columns of the basis of the domain:
T(3,1)=(10,4,7): first column. T(4,1)=(13,5,7): second column.
Is this correct?
The columns of a transformation matrix are the images of the basis vectors. Your approach is correct. Note that if the basis of the codomain were not the standard basis, you’d have to express the two images in coordinates relative to that basis.