To determine the vector $x$ in the standard basis, given the change of basis matrix $P_{C \leftarrow B}$ and the coordinate vector of $x$ with respect to basis $B$, we use the following relationship:

$x = P_{C \leftarrow B} [x]_B$

Where:

• $P_{C \leftarrow B}$ is the change of basis matrix that transforms coordinates from basis $B$ to basis $C$,
• $[x]_B$ is the coordinate vector of $x$ with respect to basis $B$,
• $x$ is the vector in the basis $C$.

Given:

• $P_{C \leftarrow B} = \begin{bmatrix} -2 & 2 & 1 & -2 \\ 2 & 4 & -2 & 2 \\ 4 & 1 & 1 & -1 \\ -1 & 2 & -1 & -2 \end{bmatrix}$
• $[x]_B = \begin{bmatrix} 2 \\ -1 \\ -1 \\ -1 \end{bmatrix}$

Now, let's compute:

$x = P_{C \leftarrow B} \cdot [x]_B$

$x = \begin{bmatrix} -2 & 2 & 1 & -2 \\ 2 & 4 & -2 & 2 \\ 4 & 1 & 1 & -1 \\ -1 & 2 & -1 & -2 \end{bmatrix} \cdot \begin{bmatrix} 2 \\ -1 \\ -1 \\ -1 \end{bmatrix}$

Let's calculate this matrix multiplication step by step.The calculated vector $x$ is:

$x = \begin{bmatrix} -5 \\ 0 \\ 7 \\ -1 \end{bmatrix}$

However, you mentioned that this is not the correct answer. This suggests there might be some issue with the input data or the matrix computation itself. Could you verify the values for $P_{C \leftarrow B}$, $[x]_B$, or clarify any other part of the problem?

Let me know if you would like to explore further. Meanwhile, here are five related questions you could consider:

1. How is the change of basis matrix $P_{C \leftarrow B}$ constructed?
2. What steps are involved in transforming vectors between different bases?
3. How do matrix multiplication properties affect change of basis operations?
4. How would you invert a change of basis transformation to recover the original vector in basis $B$?
5. What role do linear transformations play in changing coordinate systems?

Tip: Always verify the order and dimensions of vectors and matrices in linear transformations to avoid inconsistencies.