nts) 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 Ha subspace of the vector space V? 1. Is H nonempty? 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 1 2 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 idempotent matrices A and B such that (A + B)² + (A + B).) 5 6 78 3 4 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 (Hint: to show that His not closed using a comma separated list and syntax such as 2, [[3,4], [5,6]] for the answer 2, 5 6 under scalar multiplication, it is sufficient to find a real number r and an idempotent matrix A such that (rA)² + (rA).) 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

nts) 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 Ha subspace of the vector space V? 1. Is H nonempty? 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 1 2 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 idempotent matrices A and B such that (A + B)² + (A + B).) 5 6 78 3 4 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 (Hint: to show that His not closed using a comma separated list and syntax such as 2, [[3,4], [5,6]] for the answer 2, 5 6 under scalar multiplication, it is sufficient to find a real number r and an idempotent matrix A such that (rA)² + (rA).) 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

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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