Final answer:
It is not possible to find two two-dimensional subspaces of R3 whose intersection is only the zero vector because any non-trivial two-dimensional subspaces in three-dimensional space will intersect along a line, not just at a point.
Step-by-step explanation:
The question asks if it is possible to find two two-dimensional subspaces U and V of R3 (the three-dimensional real number space) whose intersection is solely the zero vector, denoted as {0}. Since both subspaces are two-dimensional, each can be represented by a pair of basis vectors. Let's denote the basis for U as {u1, u2} and the basis for V as {v1, v2}.
If you attempt to combine these four vectors into a single set, you will have four vectors in a three-dimensional space. According to the Pigeonhole Principle, there cannot be more linearly independent vectors in a space than the dimension of that space, which implies that the set {u1, u2, v1, v2} must be linearly dependent. This means that there must exist some non-trivial linear combination of these vectors that results in the zero vector. However, as U and V are only supposed to intersect at the zero vector, it is not possible for any vector in one subspace to be expressed as a combination of vectors from the other subspace except for the zero vector itself.
The geometrical interpretation of this conclusion is that any two planes in three-dimensional space (which are the geometrical representation of the two-dimensional subspaces of R3) must intersect along a line or coincide entirely, unless one of the planes is the trivial subspace {0}. Therefore, it is not possible for two two-dimensional subspaces U and V of R3 to intersect only at {0}.