Question

Consider the ordered bases B = ((1, −2), (2, −3)) and C = ((−2, 1), (−3, −1)) for the vector space R²

a. Find the transition matrix from C to the standard ordered basis E = ((1, 0), (0, 1)).
b. Find the transition matrix from В to E. TᴱB = [ - ]
C. Find the transition matrix from E to B Tᴮₑ = [ - ]
d. Find the transition matrix from C to B TB = []
e. Find the coordinates of u = (3,-2) in the ordered basis B Note that [u] = T [ul lu│B = [t]. Find the coordinates of v in the ordered basis B ir the coordinale vector of v in C is |v|C = (−1, 2)

Answer 1

To find the transition matrix from C to the standard ordered basis E, we take the inverse of matrix C and multiply it by the matrix representing the standard ordered basis E.

Step-by-step explanation:

To find the transition matrix from C to the standard ordered basis E, we can use the formula: Tᴇc = [C]ₑ[C]c⁻¹, where [C]e is the matrix representing the standard ordered basis E and [C]c⁻¹ is the inverse of the matrix representing the basis C.

For the given bases C = ((−2, 1), (−3, −1)) and E = ((1, 0), (0, 1)), we can calculate [C]c⁻¹ by taking the inverse of C. The inverse of a 2x2 matrix is given by: [A]⁻¹ = 1/(ad - bc) * [[d, -b],[-c, a]]. By substituting the values from matrix C, we get [C]c⁻¹ = [[1, -1],[3, -2]].

Then, using the formula Tᴇc = [C]ₑ[C]c⁻¹, we can calculate the transition matrix from C to E as Tᴇc = ((1, 0), (0, 1)) * [[1, -1],[3, -2]] = [[1, -1],[3, -2]].