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To solve the problem using the rank-nullity theorem, let's first recall the theorem:

**Rank-Nullity Theorem**: For a linear transformation \( T: U \to V \), the theorem states:
\[
\text{dim}(U) = \text{rank}(T) + \text{nullity}(T)
\]

Let's apply this theorem to each case in the table:

### Case 1: \( U = \mathbb{R}^4 \)

- \(\text{dim}(U) = 4\)
- \(\text{rank}(T) = 3\)
- \(\text{nullity}(T) = \text{Ex. 5}\)

Using the rank-nullity theorem:
\[
4 = 3 + \text{nullity}(T)
\]
\[
\text{nullity}(T) = 4 - 3 = 1
\]

So, \(\text{nullity}(T) = 1\).

### Case 2: \( U = \mathbb{R}^9 \)

- \(\text{dim}(U) = 9\)
- \(\text{rank}(T) = \text{Ex. 5}\)
- \(\text{nullity}(T) = 6\)

Using the rank-nullity theorem:
\[
9 = \text{rank}(T) + 6
\]
\[
\text{rank}(T) = 9 - 6 = 3
\]

So, \(\text{rank}(T) = 3\).

### Case 3: \( U = \mathbb{R}^n+2 \)

- \(\text{dim}(U) = n+2\)
- \(\text{rank}(T) = n+2\)
- \(\text{nullity}(T) = \text{Ex. 5}\)

Using the rank-nullity theorem:
\[
n+2 = (n+2) + \text{nullity}(T)
\]
\[
\text{nullity}(T) = 0
\]

So, \(\text{nullity}(T) = 0\).

### Summary of the Table

| \( U \)       | \( \mathbb{R}^4 \) | \( \mathbb{R}^9 \) | \( \mathbb{R}^{n+2} \) |
|---------------|--------------------|--------------------|------------------------|
| \( \text{dim}(U) \) | 4                  | 9                  | \( n+2 \)                |
| \( \text{rank}(T) \) | 3                  | 3                  | \( n+2 \)                |
| \( \text{nullity}(T) \) | 1                  | 6                  | 0                      |

Would you like more details or have any other questions?

Here are 5 questions that expand on this topic:

1. How is the rank-nullity theorem useful in understanding the structure of linear transformations?
2. What happens if the nullity of a transformation is zero? What does that tell us about the transformation?
3. Can the rank of a linear transformation be greater than the dimension of the codomain \( V \)?
4. How would you interpret a linear transformation with high nullity in terms of its impact on the space \( U \)?
5. What role does the dimension of the codomain \( V \) play in determining the properties of \( T \)?

**Tip:** Remember that the rank of a transformation gives us the dimension of the image (range) of the transformation, while the nullity tells us the dimension of the kernel (the set of vectors that map to zero).

Explore how to apply the Rank-Nullity Theorem to solve linear transformation problems. Learn about dimensions of vector spaces, ranks of linear transformations, and nullity calculations with detailed examples. Suitable for advanced undergraduate students and anyone interested in linear algebra concepts.

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