The trace of a square n x n matrix A = (aij) is the sum a₁1 + a₂2 + ... + ann 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? H is nonempty 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 list and syntax such as [[1,2],[3,4]], [[5,6],[7,8]] for the answer (Hint: to show that H is not closed under addition, it is sufficient to find two [36] trace zero matrices A and B such that A + B has nonzero trace.) CLOSED 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a matrix in H whose product is not in H, using a comma separated list and syntax such as 2, [[3,4],[5,6]] for the answer 2, (Hint: to show that H is not closed under scalar multiplication, it 3 4 6 is sufficient to find a real number r and a trace zero matrix A such that rA has nonzero trace.) CLOSED 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. H is a subspace of V

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