https://file.aiquickdraw.com/mathbot/file/1728493678473wfjv1l94.jpg

Let's go through each of the statements in the image step by step and determine whether they are true or false.

1. **Statement:**
   - If \( S = \text{span}(u_1, u_2, u_3) \), then \( \dim(S) = 3 \).
   - **True**. The dimension of a span of three linearly independent vectors \( u_1, u_2, u_3 \) is 3, assuming the vectors are linearly independent. If they are not linearly independent, the dimension would be less than 3, but generally, this statement assumes independence.

2. **Statement:**
   - If the set of vectors \( U \) spans a subspace \( S \), then vectors can be added to \( U \) to create a basis for \( S \).
   - **False**. If \( U \) already spans the subspace \( S \), then \( U \) already includes enough vectors to form a basis (assuming no linear dependence among the vectors in \( U \)). Therefore, no additional vectors are needed. If \( U \) spans \( S \), \( U \) is already a basis, or it can be reduced to one by removing dependent vectors.

3. **Statement:**
   - If \( S_1 \) and \( S_2 \) are subspaces of \( \mathbb{R}^n \) of the same dimension, then \( S_1 = S_2 \).
   - **False**. Two subspaces can have the same dimension without being the same subspace. For instance, two different planes in \( \mathbb{R}^3 \) could both have dimension 2, but they do not have to be the same plane.

4. **Statement:**
   - If the set of vectors \( U \) spans a subspace \( S \) but is not a basis, then vectors can be removed from \( U \) to create a basis for \( S \).
   - **True**. If \( U \) spans \( S \) but is not a basis, it means that there are some linearly dependent vectors in \( U \). By removing those dependent vectors, you can reduce \( U \) to a basis for \( S \).

5. **Statement:**
   - If the set of vectors \( U \) is linearly independent in a subspace \( S \) but is not a basis, then vectors can be added to \( U \) to create a basis for \( S \).
   - **True**. If \( U \) is linearly independent but does not span the entire subspace \( S \), then more vectors must be added to \( U \) to span \( S \), thus forming a basis.

### Summary of Answers:
1. True
2. False
3. False
4. True
5. True

Would you like further explanation on any of these points?

### Related Questions:
1. What is the formal definition of a subspace?
2. How do you determine if a set of vectors is linearly independent?
3. Can two different subspaces in \( \mathbb{R}^n \) have the same dimension but not intersect?
4. How can you construct a basis from a spanning set?
5. What is the relationship between the rank of a matrix and the dimension of its column space?

### Tip:
A set of vectors forms a basis for a subspace if and only if the vectors are both linearly independent and span the subspace.

Are the following statements true or false?
1. If S = span(u_1, u_2, u_3), then dim(S) = 3.
2. If the set of vectors U spans a subspace S, then vectors can be added to U to create a basis for S.
3. If S_1 and S_2 are subspaces of R^n of the same dimension, then S_1 = S_2.
4. If the set of vectors U spans a subspace S but is not a basis, then vectors can be removed from U to create a basis for S.
5. If the set of vectors U is linearly independent in a subspace S but is not a basis, then vectors can be added to U to create a basis for S.

This problem explores concepts in linear algebra including subspaces, bases, and linear independence. Learn how to determine the dimension of a subspace, and how to form a basis from spanning sets or linearly independent vectors. Suitable for undergraduate students.

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