Reference no: EM131396988

1. Recall the definition of a skew-symmetric n×n matrix from Homework1: a matrix A is skew-symmetric if A = -At. Then the subset Wn ⊂Mn×n(R) of skew-symmetric n×n real matrices is a real vector space (asyou already knew if you answered that homework question correctly).In particular, consider the subspace W3 ⊂ M3×3(R) of skew-symmetric3 × 3 matrices.(a) Find a basis for W3(b) Use your basis to find the dimension of W37. Suppose S is a subset of a vector space V .(a) Show that if v ∈ V is contained in span(S), then span(S ∪ {v}) =span(S).(b) Show that if v ∈ V is not contained in span(S), thendim(span(S ∪ {v})) = dim(span(S)) + 1

2. Suppose S is a subset of a vector space V .(a) Show that if v ∈ V is contained in span(S), then span(S ∪ {v}) =span(S).(b) Show that if v ∈ V is not contained in span(S), thendim(span(S ∪ {v})) = dim(span(S)) + 1.