Got it! If you're considering four vectors in $\mathbb{R}^3$, there's an important concept to keep in mind: the space you're working with is $\mathbb{R}^3$, which is three-dimensional. This means that any set of vectors you pick can span at most a three-dimensional space (i.e., $\mathbb{R}^3$).

If you're using four vectors, they can't span a space larger than $\mathbb{R}^3$, because the maximum number of linearly independent vectors in $\mathbb{R}^3$ is 3. However, those four vectors might be linearly dependent (meaning some of them can be written as a linear combination of the others), so they will not increase the dimension of the span beyond 3.

Would you like to explore how to determine if those four vectors are linearly independent, or if you want to find the span of the vectors? Feel free to share the vectors if you’d like help with that!

Here are some questions for further exploration:

1. How do you check if a set of vectors is linearly independent in $\mathbb{R}^3$?
2. What does it mean for vectors to span a subspace of $\mathbb{R}^3$?
3. How can you find the dimension of the span of a set of vectors?
4. How do you express the span of a set of vectors in $\mathbb{R}^3$?
5. Can four vectors ever span the entire space of $\mathbb{R}^3$?

Tip: If you're trying to figure out the linear dependence or independence of your vectors, setting up the matrix of these vectors and performing row reduction (Gaussian elimination) can help you easily determine this!