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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 [1 2] E 1E 1: (Hint: to show that H is not closed under addition, it is sufficient to find two [5 6] answer idempotent matrices A and B such that (A + B)² + (A+B).) [1.0].[0,0].[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 [3 4] 2, [[3,4], [5,6]] for the answer 2, : d: (Hint: to show that H is not closed under scalar [5 multiplication, it is sufficient to find a real number r and an idempotent matrix A such that (rA)² + (rA).) 2([1,0].[0,0]) 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 v O

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 [1 2] E 1E 1: (Hint: to show that H is not closed under addition, it is sufficient to find two [5 6] answer idempotent matrices A and B such that (A + B)² + (A+B).) [1.0].[0,0].[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 [3 4] 2, [[3,4], [5,6]] for the answer 2, : d: (Hint: to show that H is not closed under scalar [5 multiplication, it is sufficient to find a real number r and an idempotent matrix A such that (rA)² + (rA).) 2([1,0].[0,0]) 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 v O

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CR: Review Exercises
Problem 41CR: Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the...
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A square matrix A is idempotent if A? = 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 [1 2] [5 6] 3 E 1E : (Hint: to show that H is not closed under addition, it is sufficient to find two answer 7 8 idempotent matrices A and B such that (A + B)? + (A + B).) [1.0].[0,0].[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 [3 4] 2, [[3,4], [5,6]] for the answer 2, :1. (Hint: to show that H is not closed under scalar 5 6 multiplication, it is sufficient to find a real number r and an idempotent matrix A such that (rA)² + (rA).) 2([1,0].[0,0]) 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 ♥ O
Transcribed Image Text:A square matrix A is idempotent if A? = 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 [1 2] [5 6] 3 E 1E : (Hint: to show that H is not closed under addition, it is sufficient to find two answer 7 8 idempotent matrices A and B such that (A + B)? + (A + B).) [1.0].[0,0].[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 [3 4] 2, [[3,4], [5,6]] for the answer 2, :1. (Hint: to show that H is not closed under scalar 5 6 multiplication, it is sufficient to find a real number r and an idempotent matrix A such that (rA)² + (rA).) 2([1,0].[0,0]) 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 ♥ O
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