Let v m22 let w be the collection of all 2 2 symmetric

Question

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.