b) Let V = M 2 2. Let W be the collection of all 2 2 symmetric matrices and U be the set of all 2 2 skew-symmetric matrices.

b)    Let V = M2×2. Let W be the collection of all 2 × 2 symmetric matrices and U be the set of all 2 × 2 skew-symmetric matrices.

i)      Assuming that W is a subspace of V , find a basis for W and thereby determine the dimension of W.

ii)    Assuming that U is a subspace of V , find a basis for U and hence determine its dimension. iii) Prove that if A ∈ V , then A = B +C with some B ∈ W and C ∈ U.