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.