Question

Consider the following two ordered bases of R³:

B = { [ 0,1,1 ], [ 0,-2,-1 ], [ -1,-3,-2 ] }
C = { [ -2,-1,-1 ], [ 1,0,1 ], [ 1,3,2 ] }
Find the transition matrix from B to C.

Answer 1

To find the transition matrix from base B to base C, calculate the inverse of the matrix formed by base C and multiply it by the matrix formed by base B.

Step-by-step explanation:

To find the transition matrix from the ordered base B to the ordered base C, we can use the equation PC→B = C-1 × B, where PC→B is the transition matrix we are looking for, C-1 is the inverse of the matrix formed by the vectors in base C, and B is the matrix formed by the vectors in base B. To solve this, we perform the following steps:

1. Create matrices C and B with the given vectors.
2. Find the inverse matrix C-1 using appropriate methods such as Gauss-Jordan elimination or adjugate and determinant.
3. Multiply C-1 by B to get the transition matrix PC→B.

Performing these calculations involves knowledge of linear algebra techniques more than what is typically taught in high school, thus indicating that this is a college-level mathematics problem.