Question

(1 point) A square matrix A is idempotent if A2 = A. 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 idempotent matrices with real entries. Is H a subspace of the vector space V? 1. Does H contain the zero vector of V? choose 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 1 for the newer [1 2] [5 6] Wint-to show . (Hint: to show that H is not closed under addition, it is sufficient to find two ver 3 4) '17 8: idempotent matrices A and B such that (A + B)2 + (A + B).) 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 3 41 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 is sufficient 5 6 to find a real number r and an idempotent matrix A such that (TA) + (TA).) 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

Answer 1

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