Final answer:
The first and third statements are false, correcting them: a linearly independent set spanning a subspace is already a basis, and three vectors lying in the same plane cannot form a basis for R3. The second statement is true; a spanning set can be reduced to a basis.
Step-by-step explanation:
The statement 'If the set of vectors U is linearly independent in subspace S then vectors can be removed from U to create basis for S' is false. A correct statement would be: If the set of vectors U spans subspace S and is linearly dependent, then vectors can be removed from U to create a basis for S. A linearly independent set that spans a subspace is already a basis for that subspace and does not need to have vectors removed.
Regarding 'If the set of vectors U spans subspace S, then vectors can be removed from U to create basis for S', the statement is true. If a set spans a subspace but is not linearly independent, we can remove vectors from it until we end up with a minimal spanning set that is also linearly independent, which is the definition of a basis for the subspace.
Concerning the idea that 'Three nonzero vectors that lie in a plane in R3 might form a basis for R3', this is false. The correct statement would be: at most, two nonzero vectors that lie in a plane in R3 can form a basis for that plane. Three vectors are needed to span R3, and if they all lie in the same plane, they are linearly dependent and cannot form a basis for R3.