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 2idempotent matrices with real entries. Is H a subspace of the vector space V?1. Is H nonempty?choose2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in H whosesum is not in H, using a comma separated list and syntax such as1 2[[1,2], [3,4]], [[5,6], [7,8]] for the answer5 6(Hint: to show83 4 7that H is not closed under addition, it is sufficient to find two idempotent matrices A and Bsuch that (A + B)² ‡ (A + B).)23. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in Rand a matrix in H whose product is not in H, using a comma separated list and syntax such3 4as 2, [[3,4], [5,6]] for the answer 2,(Hint: to show that H is not closed6under scalar multiplication, it is sufficient to find a real number r and an idempotent matrix Asuch that (rA)2 # (rA).)4. Is H a subspace of the vector space V? You should be able to justify your answer by writinga complete, coherent, and detailed proof based on your answers to parts 1-3.choose

Answered: Image /qna-images/answer/801d14c3-317c-4179-af61-c06bb35d17ba.jpg

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. 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 list and syntax such as 1 2 [[1,2], [3,4]], [[5,6], [7,8]] for the answer CO ∞ 2 6 5 (Hint: to show 3 4 7 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).) 8 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 3 4 as 2, [[3,4], [5,6]] for the answer 2, (Hint: to show that H is not closed 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

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. 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 list and syntax such as 1 2 [[1,2], [3,4]], [[5,6], [7,8]] for the answer CO ∞ 2 6 5 (Hint: to show 3 4 7 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).) 8 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 3 4 as 2, [[3,4], [5,6]] for the answer 2, (Hint: to show that H is not closed 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

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section: Chapter Questions

Problem 8RQ

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