Consider the ordered bases
B = {2 - 9x, 1 - 4x}
C = {3x - 2, 3x - 3}
for the vector space P2
E = {1, x}
Find the transition matrix from C to B
= ???
To find the transition matrix from C to B, we need to express the basis vectors of B in terms of the basis vectors of C.
First, let's express the vector (2 - 9x) in terms of the basis vectors of C:
(2 - 9x) = a(3x - 2) + b(3x - 3)
Expanding the right side, we get:
2 - 9x = (3ax - 2a) + (3bx - 3b)
Comparing the coefficients of the same powers of x, we get two equations:
3a + 3b = 0 (for the constant term)
-2a - 3b = 2 (for the x term)
Solving this system of equations, we find a = -2/3 and b = 2/3.
Therefore, we can express (2 - 9x) as:
(2 - 9x) = (-2/3)(3x - 2) + (2/3)(3x - 3)
Similarly, we can express (1 - 4x) in terms of the basis vectors of C:
(1 - 4x) = c(3x - 2) + d(3x - 3)
Expanding the right side, we get:
1 - 4x = (3cx - 2c) + (3dx - 3d)
Comparing the coefficients of the same powers of x, we get two equations:
3c + 3d = 0 (for the constant term)
-2c - 3d = -4 (for the x term)
Solving this system of equations, we find c = -2/9 and d = -2/9.
Therefore, we can express (1 - 4x) as:
(1 - 4x) = (-2/9)(3x - 2) + (-2/9)(3x - 3)
Now, we can construct the transition matrix. The columns of the matrix will be the coefficients of the basis vectors of B expressed in terms of the basis vectors of C.
The transition matrix from C to B is:
[ -2/3 -2/9 ]
[ 2/3 -2/9 ]