Question

Find the transition matrix from basis B to basis B' when: B={(−1,0,0),(0,1,0),(0,0,−1)} and B' ={(0,0,4),(1,5,0),(6,0,4)} Please proceed with the calculations and steps to determine the transition matrix.

Answer 1

To find the transition matrix from basis B to basis B', we express each vector of B' as a linear combination of the vectors in B and arrange the corresponding coefficients into a matrix, resulting in the matrix [B to B'] = | 0 -1 -6 | | 0 5 0 | | -4 0 -4 |.

Step-by-step explanation:

To find the transition matrix from basis B to basis B', we need to express each vector of basis B' in terms of the basis B. First, we write each vector of B' as a linear combination of the vectors in B. For ease of calculation, we can arrange the vectors of B as columns in a matrix called the change of basis matrix and perform linear algebra operations to find the corresponding coefficients that express vectors of B' in terms of B.

Performing this calculation, we start with the original bases:

• B = {(-1,0,0), (0,1,0), (0,0,-1)}
• B' = {(0,0,4), (1,5,0), (6,0,4)}

We form a matrix (B) with the vectors of B as its columns:

B = | -1 0 0 |
| 0 1 0 |
| 0 0 -1 |

Next, we write each vector of B' as a linear combination of the columns of matrix B:

• (0,0,4) = 0*(-1,0,0) + 0*(0,1,0) - 4*(0,0,-1)
• (1,5,0) = -1*(-1,0,0) + 5*(0,1,0) + 0*(0,0,-1)
• (6,0,4) = -6*(-1,0,0) + 0*(0,1,0) - 4*(0,0,-1)

The coefficients form the transition matrix as follows:

Transition Matrix [B to B'] = | 0 -1 -6 |
| 0 5 0 |
| -4 0 -4 |

This matrix is obtained by placing the coefficients that correspond to the vectors of B' in the columns of the new matrix relative to the original basis B.