Let's go through each statement to determine if it is true or false:

1. If $S = \text{span} \{ u_1, u_2, u_3 \}$, then $\dim(S) = 3$.

   • False, unless we know that $u_1, u_2,$ and $u_3$ are linearly independent. The span of three vectors will only have a dimension of 3 if those vectors are linearly independent. If they are not, the dimension of $S$ could be less than 3.

2. If the set of vectors $U$ is linearly independent in a subspace $S$, then vectors can be removed from $U$ to create a basis for $S$.

   • True. If $U$ is linearly independent, it can serve as a potential basis or a part of a basis for $S$. Removing vectors from $U$ could yield a basis for $S$ as long as the remaining vectors still span $S$.

3. 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 identical. For instance, two different planes through the origin in $\mathbb{R}^3$ both have dimension 2, but they are not the same plane.

4. If the set of vectors $U$ is linearly independent in a subspace $S$, then vectors can be added to $U$ to create a basis for $S$.

   • True. If $U$ is linearly independent but does not yet span $S$, then additional vectors can be added to complete a basis for $S$, thus spanning the entire subspace.

5. If the set of vectors $U$ spans a subspace $S$, then vectors can be removed from $U$ to create a basis for $S$.

   • True. If $U$ spans $S$, it might contain redundant vectors. Removing some of these while maintaining linear independence will yield a basis for $S$.

If you'd like more details or have any questions, let me know.

Here are five related questions to expand on these concepts:

1. What is the difference between spanning a subspace and forming a basis for it?
2. How do we determine the dimension of a subspace given a set of vectors?
3. Can two subspaces with the same dimension intersect in a specific way?
4. How can we check if a set of vectors in a subspace is linearly independent?
5. Why is linear independence important when forming a basis?

Tip: A set of vectors spans a subspace if every vector in the subspace can be written as a linear combination of the set's vectors, while a basis requires the set to be both spanning and linearly independent.