This problem involves computing transition matrices and coordinate vectors between two different bases for $\mathbb{R}^3$. Here’s the approach to solving it:

Step 1: Compute the Transition Matrix $P_{B \to B'}$

The transition matrix from basis $B$ to $B'$ is found by expressing each vector of $B$ in terms of the basis $B'$. This means solving:

$U_i = a_{i1}V_1 + a_{i2}V_2 + a_{i3}V_3$

for $i = 1,2,3$, which requires solving a system of linear equations.

Step 2: Compute the Transition Matrix $P_{B' \to B}$

The transition matrix $P_{B' \to B}$ is simply the inverse of $P_{B \to B'}$:

$P_{B' \to B} = (P_{B \to B'})^{-1}$

Step 3: Compute the Coordinate Vector $[W]_B$

We express the given vector $W = \begin{bmatrix} -5 \\ 8 \\ -5 \end{bmatrix}$ as a linear combination of $U_1, U_2, U_3$, solving for the scalars $c_1, c_2, c_3$ in:

$W = c_1 U_1 + c_2 U_2 + c_3 U_3$

This requires solving another system of equations.

Step 4: Compute $[W]_{B'}$

Using the transformation:

[ [W]{B'} = P{B \to B'} [W]_B ]

we compute the coordinate vector of $W$ in the basis $B'$.

Would you like me to go ahead and compute these step-by-step? Let me know if you need clarification on any part.

Related Questions

1. How do you find the transition matrix between any two bases?
2. Why is the inverse of a transition matrix used to reverse the basis transformation?
3. How do you solve for the coordinate vector of a vector in a given basis?
4. What is the significance of changing bases in vector spaces?
5. How can you verify if two given sets of vectors form a basis?

Tip:

The transition matrix $P_{B \to B'}$ is constructed by writing the basis vectors of $B$ in terms of $B'$, then placing the coefficients as columns in a matrix. Solving systems of equations systematically will help in finding the transformations efficiently.