Matrix of Linear Transformation

Question

Find a matrix for the Linear Transformation T: R2 → R3, defined by T (x, y) = (13x - 9y, -x - 2y, -11x - 6y) with respect to the basis B = {(2, 3), (-3, -4)} and C = {(-1, 2, 2), (-4, 1, 3), (1, -1, -1)} for R2 & R3 respectively.

Here, the process should be to find the transformation for the vectors of B and express those as a linear combination of C and those vectors will form the matrix for linear transformation. Is my approach correct or do I need to change something?

Answer 1

I will show you how to do in python with sympy module

import sympy
# Assuming that B(x, y) = (2,3)*x + (-3, -4)*y it can be expressed as a left multiplication by
B = sympy.Matrix(
    [[2, -3],
     [3, -4]])

# Then you apply T as a left multiplication by
T = sympy.Matrix(
    [[13, -9],
     [-1, -2],
     [-11, -6]])

#And finally to get the representation on the basis C you multiply of the result
# by the inverse of C

C = sympy.Matrix(
    [[-1, -4, 1],
     [2, 1, -1],
     [2, 3, -1]])

combined = C.inv() * T * B

The combined transformation matrix yelds

[[-57, 77], 
 [-16, 23], 
 [-122, 166]])