Let's review and evaluate the given statements individually:

1. False:
   This statement is indeed false. Two subspaces $S_1$ and $S_2$ of $\mathbb{R}^n$ can have the same dimension but still be different. For example, different lines through the origin in $\mathbb{R}^2$ can both have dimension 1 but are distinct subspaces.

2. True:
   This statement is true. If $S = \text{span}\{u_1, u_2, u_3\}$ and the vectors $u_1, u_2, u_3$ are linearly independent, then the dimension of the span (subspace $S$) is equal to the number of independent vectors, which is 3.

3. True:
   This is true. If a set of vectors $U$ is linearly independent in a subspace $S$, you can always add more vectors (if necessary) to $U$ to form a basis for $S$, thereby completing the spanning set.

4. False:
   This statement is false. If the set of vectors $U$ already spans the subspace $S$, then adding more vectors to $U$ will not create a basis. In fact, it will introduce redundancy and linear dependence. A basis for $S$ requires only the minimal set of linearly independent vectors that span $S$.

5. True:
   This statement is true. If a set of vectors $U$ spans a subspace $S$, you can remove redundant vectors (i.e., linearly dependent vectors) from $U$ to create a basis for $S$, which is the smallest set of linearly independent vectors that span the space.

Would you like further clarification on any of these explanations?

Here are 5 related questions to expand your understanding:

1. How do you determine if a set of vectors is linearly independent?
2. What is the difference between a spanning set and a basis?
3. Can two subspaces of different dimensions in $\mathbb{R}^n$ be equal?
4. Why is linear dependence important when constructing a basis?
5. How do you find the dimension of a subspace given a set of vectors?

Tip: When building a basis, focus on identifying the linearly independent vectors—this ensures that you span the space without unnecessary redundancy.