Find the transition matrix from B to B'.B = {(−1, 0, 0), (0, 1, 0), (0, 0, −1)}, B' = {(0, 0, 5), (1, 2, 0), (7, 0, 5)}

To find the transition matrix from B to B', we need to express each vector in B' as a linear combination of the vectors in B.

Let's denote the vectors in B as b1 = (-1, 0, 0), b2 = (0, 1, 0), and b3 = (0, 0, -1). Similarly, let's denote the vectors in B' as b1' = (0, 0, 5), b2' = (1, 2, 0), and b3' = (7, 0, 5).

We need to solve the following systems of equations:

1. b1' = a11b1 + a21b2 + a31*b3
2. b2' = a12b1 + a22b2 + a32*b3
3. b3' = a13b1 + a23b2 + a33*b3

Solving these systems, we get:

1. 0 = -a11 + 0 + 0 and 0 = 0 + a21 + 0 and 5 = 0 + 0 - a31, which gives a11 = 0, a21 = 0, and a31 = -5.
2. 1 = -a12 + 0 + 0 and 2 = 0 + a22 + 0 and 0 = 0 + 0 - a32, which gives a12 = -1, a22 = 2, and a32 = 0.
3. 7 = -a13 + 0 + 0 and 0 = 0 + a23 + 0 and 5 = 0 + 0 - a33, which gives a13 = -7, a23 = 0, and a33 = -5.

Therefore, the transition matrix from B to B' is:

| 0 -1 -7 |
| 0  2  0 |
|-5  0 -5 |