Final answer:
The space spanned by two vectors in R^3 is a line if one vector is a scalar multiple of the other, a plane if they are linearly independent, or simply the origin if both vectors are the zero vector.
Step-by-step explanation:
To prove that the space spanned by two vectors in R^3 (three-dimensional space) is either a line through the origin, a plane through the origin, or the origin itself, we need to consider different scenarios based on the vectors' relationship.
If both vectors are the zero vector, they only span the origin itself. If one vector is a nonzero scalar multiple of the other, then they span a line through the origin since any linear combination of the two will still lie on that line.
When the vectors are linearly independent (not scalar multiples of each other), they will span a plane through the origin. Any linear combination of the two vectors will form a vector that lies within this plane because the definition of a plane in R^3 includes all linear combinations of any two non-collinear vectors.
In R^3, a vector Ả has the form Ả = AxÎ + AyĴ + A₂Î, representing its components along the x, y, and z-axes, defined by the unit vectors î, ĵ, and k, respectively. Thus, when we have two such vectors, the combinations of these components determine the geometry of the space they span.