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To solve the problem using the rank-nullity theorem, we need to recall the relationship:

\[
\text{dim}(U) = \text{rank}(T) + \text{nullity}(T)
\]

This theorem states that for a linear transformation \( T: U \to V \), the dimension of the domain \( U \) is equal to the sum of the rank of the transformation and the nullity of the transformation.

Given the table:

| \( U \)    | \( P_6 \) | \( P_7 \) | \( P_n \) |
|------------|-----------|-----------|-----------|
| \(\text{dim}(U)\)   | 6         | ?         | ?         |
| \(\text{rank}(T)\)  | Ex: 5     | 8         | Ex: n+2   |
| \(\text{nullity}(T)\) | 1         | Ex: 5     | 7         |

We'll solve each case using the formula:

1. **For \( P_6 \):**
   - Given: \(\text{dim}(U) = 6\), \(\text{nullity}(T) = 1\).
   - Using the formula: \(\text{dim}(U) = \text{rank}(T) + \text{nullity}(T)\), we get:
     \[
     6 = \text{rank}(T) + 1
     \]
     Therefore, \(\text{rank}(T) = 5\).

2. **For \( P_7 \):**
   - Given: \(\text{rank}(T) = 8\), and the missing values for \(\text{dim}(U)\) and \(\text{nullity}(T)\) are to be calculated.
   - Using the same formula:
     \[
     \text{dim}(U) = 8 + \text{nullity}(T)
     \]
   - Since the table suggests \(\text{nullity}(T) = 5\), then:
     \[
     \text{dim}(U) = 8 + 5 = 13
     \]

3. **For \( P_n \):**
   - Given: \(\text{nullity}(T) = 7\), and \(\text{rank}(T) = n + 2\).
   - Using the formula:
     \[
     \text{dim}(U) = (n + 2) + 7 = n + 9
     \]

Now, let's update the table:

| \( U \)    | \( P_6 \) | \( P_7 \) | \( P_n \) |
|------------|-----------|-----------|-----------|
| \(\text{dim}(U)\)   | 6         | 13        | \(n + 9\)  |
| \(\text{rank}(T)\)  | 5         | 8         | \(n + 2\)  |
| \(\text{nullity}(T)\) | 1         | 5         | 7         |

Would you like any further details or explanations? 

Here are some related questions:
1. How does the rank-nullity theorem apply to different types of vector spaces?
2. What are some examples where the rank-nullity theorem is used in real-world applications?
3. How does changing the dimensions of the domain \( U \) affect the rank and nullity of \( T \)?
4. Can the rank of a transformation exceed the dimension of the domain?
5. What happens to the nullity if the transformation is injective?

**Tip:** Always ensure that the sum of the rank and nullity matches the dimension of the domain when applying the rank-nullity theorem.

Understand how to apply the Rank-Nullity Theorem in linear transformations. Solve specific problems involving the dimensions of domains, ranks, and nullities using the theorem. Detailed examples and explanations provided.

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