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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? H contains the zero vector of V Ⓒ 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 idempotent matrices A and B such 4] [7 that (A + B)2 (A + B).) ([1,0], [0,1]).([1,0], [0,1]) 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, [1 . (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r and an idempotent matrix A such that (rA)2 = (r.A).) ([1,0], [0,1]) 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 not a subspace of V

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? H contains the zero vector of V Ⓒ 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 idempotent matrices A and B such 4] [7 that (A + B)2 (A + B).) ([1,0], [0,1]).([1,0], [0,1]) 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, [1 . (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r and an idempotent matrix A such that (rA)2 = (r.A).) ([1,0], [0,1]) 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 not a subspace of V

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section: Chapter Questions
Problem 1RQ
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A square matrix A is idempotent if A² = A. Let V be the vector space of all 2 × 2 matrices with real entries. Let H be the set of all 2 × 2 idempotent matrices with real entries. Is H a subspace of the vector space V? 1. Does H contain the zero vector of V? H contains the zero vector of V 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 [16] (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).) ([1,0], [0,1]), ([1,0],[0,1]) 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 [3 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 to find a real number r and an idempotent matrix 5 A such that (rA)2 ‡ (rA).) ([1,0], [0,1]) 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 not a subspace of V
Transcribed Image Text:A square matrix A is idempotent if A² = A. Let V be the vector space of all 2 × 2 matrices with real entries. Let H be the set of all 2 × 2 idempotent matrices with real entries. Is H a subspace of the vector space V? 1. Does H contain the zero vector of V? H contains the zero vector of V 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 [16] (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).) ([1,0], [0,1]), ([1,0],[0,1]) 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 [3 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 to find a real number r and an idempotent matrix 5 A such that (rA)2 ‡ (rA).) ([1,0], [0,1]) 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 not a subspace of V
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