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The problem presented is about vector spaces over a field \( F \), with a focus on changing between different bases and determining the relationship between transformation matrices. Let’s tackle each part of the question systematically.

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**Given:**
- Vector space \( V \) over the field \( F \), with dimension 3.
- Basis \( B = \{v_1, v_2, v_3\} \).
  
We are asked to:
1. Find specific vectors in terms of the basis \( B \).
2. Use a new basis \( C = \{v_1, v_1 + 4v_3, v_3 - v_2\} \), and find the matrix \( P \) that changes coordinates from basis \( B \) to basis \( C \).
3. Find matrix \( Q \) such that \( Q(v)_C = (v)_B \), and determine the relationship between \( P \) and \( Q \).

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### **Part a):**
We are asked to find:

\[
(v_1)_B, (v_1 + 4v_3)_B, (v_3 - v_2)_B
\]

Since the problem explicitly states that these vectors are to be expressed in the standard basis \( B = \{v_1, v_2, v_3\} \), we will express each in terms of the basis:

- \( (v_1)_B \) simply refers to the coordinates of \( v_1 \) in basis \( B \), which is \( [1, 0, 0]^T \) because \( v_1 \) is itself a basis vector.
  
- \( (v_1 + 4v_3)_B \) is the linear combination of \( v_1 \) and \( v_3 \), so in basis \( B \), it will be:

\[
(v_1 + 4v_3)_B = [1, 0, 4]^T
\]

- \( (v_3 - v_2)_B \) is the linear combination of \( v_3 \) and \( v_2 \), which gives:

\[
(v_3 - v_2)_B = [0, -1, 1]^T
\]

So, the answers for part (a) are:

\[
(v_1)_B = [1, 0, 0]^T, \quad (v_1 + 4v_3)_B = [1, 0, 4]^T, \quad (v_3 - v_2)_B = [0, -1, 1]^T
\]

---

### **Part b):**
We are now working with a new basis \( C = \{v_1, v_1 + 4v_3, v_3 - v_2\} \), and we need to find the matrix \( P \) such that for any vector \( v \in V \):

\[
P(v)_B = (v)_C
\]

Here, the matrix \( P \) will be the change of basis matrix from \( B \) to \( C \). Each column of \( P \) will be the coordinates of the vectors in \( C \) expressed in the basis \( B \). From part (a), we already found those coordinates:

- \( v_1 \) in basis \( B \) is \( [1, 0, 0]^T \),
- \( v_1 + 4v_3 \) in basis \( B \) is \( [1, 0, 4]^T \),
- \( v_3 - v_2 \) in basis \( B \) is \( [0, -1, 1]^T \).

Thus, the matrix \( P \) is:

\[
P = \begin{bmatrix}
1 & 1 & 0 \\
0 & 0 & -1 \\
0 & 4 & 1
\end{bmatrix}
\]

---

### **Part c):**
Next, we are asked to find matrix \( Q \) such that:

\[
Q(v)_C = (v)_B
\]

This matrix \( Q \) is the inverse of \( P \) because \( P \) transforms coordinates from basis \( B \) to \( C \), and \( Q \) does the reverse. Therefore, we compute \( Q = P^{-1} \).

Let's compute the inverse of matrix \( P \):

\[
P = \begin{bmatrix}
1 & 1 & 0 \\
0 & 0 & -1 \\
0 & 4 & 1
\end{bmatrix}
\]

We can calculate \( P^{-1} \) using standard methods (Gaussian elimination or direct formula for 3x3 matrices). 

By calculating, we get:

\[
Q = P^{-1} = \begin{bmatrix}
1 & 0 & 0 \\
-1/4 & 0 & -1 \\
0 & -1 & 0
\end{bmatrix}
\]

---

### **Conclusion:**
The relationship between \( P \) and \( Q \) is that \( Q \) is the inverse of \( P \), i.e., \( Q = P^{-1} \).

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Would you like more details or clarifications on any part of this solution?

Here are 5 related questions to consider:
1. How do you determine a change of basis matrix in general?
2. What are the properties of a transformation matrix between two different bases?
3. Why is the inverse of a change of basis matrix important?
4. How would the solution change if the vector space had a different dimension?
5. Can the matrix \( P \) ever be singular? What does that imply about the two bases?

**Tip**: Always check the invertibility of transformation matrices when switching between bases to ensure that the change of basis is well-defined.

Misal V adalah ruang vektor atas lapangan F yang berdimensi 3 dengan basis B = {v1, v2, v3}. Temukan: a) v1)_B, (v1 + 4v3)_B, (v3 - v2)_B. b) Misal C = {v1, v1 + 4v3, v3 - v2} basis terurut lain dari V, temukan matriks P sehingga P(v)_B = (v)_C untuk semua vektor v ∈ V. c) Temukan matriks Q sehingga Q(v)_C = (v)_B. Bagaimana hubungan matriks P dan matriks Q?

Explore a solution to a change of basis problem in a 3-dimensional vector space. This problem covers how to find transformation matrices P and Q and their relationship in linear algebra.

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