Final answer:
The statement is false because the intersection of two subspaces S and T is not guaranteed to have the same span as the individual spans of S and T. An example using the x-axis and y-axis in R^2 helps to illustrate the discrepancy, demonstrating why the equation [span of S∩T] = [span S] ∩ [span T] does not hold true.
Step-by-step explanation:
The statement "If S and T are subspaces and S∩T, then the [span of S∩T] = [span S] ∩ [span T]" is false. Subspaces S and T are sets that themselves satisfy all the properties of a vector space, and their intersection S∩T is also a subspace.
However, the span of a set is the set of all linear combinations of its vectors. It's possible for S and T to have different vectors such that their spans are not equal, even though they might share some vectors in their intersection. A correct statement would be "the span of S∩T is a subspace of the intersection of the spans of S and T."
A counter-example would be taking S to be the x-axis in R^2 and T to be the y-axis. Their intersection S∩T is the origin {0}, which spans a subspace consisting only of the zero vector. However, the span of S is the entire x-axis and the span of T is the entire y-axis, and their intersection as spans is still only the zero vector, so the original statement is not valid because [span S] ∩ [span T] includes all possible vectors on both axes, which is far larger than the span of their intersection.