Let's analyze each sentence and identify the correct options for the blanks.

1. First Sentence:

   • "A linearly independent subset of a subspace of $\mathbb{R}^n$ ___ be larger than a spanning set for that subspace."
   • A linearly independent subset cannot have more vectors than a spanning set since a spanning set by definition has the minimum number of vectors needed to span the space. So, the correct option is "cannot".

2. Second Sentence:

   • "A basis for a subspace is a set of vectors that is both ___ and a ___ for that subspace."
   • A basis must be both "linearly independent" and a "spanning set" for the subspace. Therefore, the correct options are "linearly independent" and "spanning set".

3. Third Sentence:

   • "___ for a subspace of $\mathbb{R}^n$ are all of the same size."
   • All bases for a subspace must have the same number of elements (the dimension of the subspace). Therefore, the correct option is "Bases".

4. Fourth Sentence:

   • "The dimension of a subspace of $\mathbb{R}^n$ is the size of any ___ for that subspace."
   • The dimension of a subspace is defined as the number of elements in any of its bases. Therefore, the correct option is "basis".

Final Answers:

1. "A linearly independent subset of a subspace of $\mathbb{R}^n$ cannot be larger than a spanning set for that subspace."
2. "A basis for a subspace is a set of vectors that is both linearly independent and a spanning set for that subspace."
3. "Bases for a subspace of $\mathbb{R}^n$ are all of the same size."
4. "The dimension of a subspace of $\mathbb{R}^n$ is the size of any basis for that subspace."

Would you like further details on any of these explanations or have any questions?

Here are some related questions to expand on this concept:

1. What is the difference between a basis and a spanning set?
2. How does the dimension of a subspace relate to its basis?
3. Can a spanning set be linearly dependent? Why or why not?
4. How would you find a basis for a subspace of $\mathbb{R}^n$ in practice?
5. What happens to the basis and dimension if we change the subspace in $\mathbb{R}^n$?

Tip: Remember, a basis is both the smallest set of vectors that spans a subspace and is linearly independent.