Final answer:
In three-dimensional space, the possible spans for a set of three vectors depend on whether the vectors are linearly independent or linearly dependent.
Step-by-step explanation:
In three-dimensional space, a set of three vectors can have a variety of possible spans. The span of a set of vectors refers to all the linear combinations of those vectors. The span is the set of all possible vectors that can be formed by multiplying each vector by different scalars and adding them together.
If the three vectors are linearly independent, meaning that no vector can be expressed as a linear combination of the other two, then the span will be the entire three-dimensional space. In other words, any vector in three dimensions can be expressed as a linear combination of these three vectors.
However, if the three vectors are linearly dependent, meaning that one vector can be expressed as a linear combination of the other two, then the span will be a plane in three-dimensional space. The span will be the set of all vectors lying on that plane.