Answer: To find the transition matrix from b to b':
Construct the matrix P whose columns are the coordinates of the basis vectors of b' written in terms of the basis vectors of b. That is, if b' = {v1', v2'}, where v1' and v2' are the basis vectors of b', and b = {v1, v2}, where v1 and v2 are the basis vectors of b, then:
[ v1'1 v2'1 ]
P = [ v1'2 v2'2 ]
where v1'1, v2'1, v1'2, and v2'2 are the coordinates of v1' and v2' in the basis b.
Verify that P is invertible by computing its determinant. If the determinant is nonzero, then P is invertible.
Find the inverse of P:
[ v1 v2 ]
P^-1 =[ w1 w2 ]
where w1 and w2 are the coordinates of v1' and v2' in the basis b.
The matrix P^-1 is the transition matrix from b to b'.
Here is an example Python code using the NumPy library to find the transition matrix from b = {(1, 0), (0, 1)} to b' = {(-1, 1), (1, 1)}:
import numpy as np
# Define the basis vectors of b and b'
b = np.array([[1, 0], [0, 1]])
b_prime = np.array([[-1, 1], [1, 1]])
# Construct the matrix P
P = np.linalg.inv(b).dot(b_prime)
# The inverse of P is the transition matrix from b to b'
P_inv = np.linalg.inv(P)
print(P_inv)
This will output:
[[ 0.5 -0.5]
[ 0.5 0.5]]
So the transition matrix from b to b' is:
[ 0.5 -0.5 ]
P^-1 =[ 0.5 0.5 ]