s) The trace of a square n x n matrix A = (a) is the sum a₁ + a22 ++ aan of the entries on its main diagonal. Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 x 2 matrices with real entries that have trace 0. Is H a subspace of the vector space V? 1. Is H nonempty? choose 5 6 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in H whose sum is not in H, using a comma separated 12 list and syntax such as [[1,2], [3,4]], [[5,6], [7,8]] for the answer (Hint: to show that H is not closed 3 7 under addition, it is sufficient to find two trace zero matrices A and B such that A + B has nonzero trace.) 3 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in IR and a matrix in H whose product is not in H 3 4 using a comma separated list and syntax such as 2, [[3,4], [5,6]] for the answer 2, (Hint: to show that is not closed 56 under scalar multiplication, it is sufficient to find a real number and a trace zero matrix A such that rA has nonzero trace.) 4. Is H a subspace of the vector space V? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3. choose