The Question:

Give an example of a subset of R² that is a nontrivial subspace of R², showing all work.

My answer: 

A subset of  R² which is a nontrivial subspace of R² may be determined when a 2x2 matrices is closed under addition and multiplication. If one or more of the matrices entries are zero or linearly correlated the matrix will fulfill the properties of closure.

The following is an example of a subspace W of V:

If the vector (5,4) and all of the scalar multiples, then a nontrivial subspace of V: W= { (0,  0),  (3,  4),  (6,  8),  (9,  12),  (12,  16),  (15,  20),  (18,  24)}. The subspace is the set of vector s{(3x, 4x)} where x is any real number. If v1 + v2
 and are two vectors in the subspace then addition and scalar multiplication may be used to check. v1 + v2 = a(3,4) + b(3,4) = (a+b)(3,4) found in W. Additionally, a*v + ab(3,4) which is also found in W and for all v which is in V. This demonstrates how W is a subspace of V, since W is closed under addition and multiplication.

Evaluation comments:

The work describes a set W that is a valid subspace. It is demonstrated that W is closed under vector addition. The statement, "a*v+ab(5,4) is unclear related to the establishment of closure for scalar multiplication. It is not made evident that W contains the zero vector for specified values of a and / or b. Also, the discussion that establishes W as being nontrivial is not evident.