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 e 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], [3,4]], [[5,6],[7,8]] for the answer [33] [58] (A + B)² (A + B).) . (Hint: to show that His not closed under addition, it is sufficient to find two idempotent matrices A and B such that 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, using a comma separated list and syntax such as [34 2, [[3,4],[5,6]] for the answer 2, 3] (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 5 (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 e 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], [3,4]], [[5,6],[7,8]] for the answer [33] [58] (A + B)² (A + B).) . (Hint: to show that His not closed under addition, it is sufficient to find two idempotent matrices A and B such that 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, using a comma separated list and syntax such as [34 2, [[3,4],[5,6]] for the answer 2, 3] (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 5 (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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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 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], [3,4]], [[5,6],[7,8]] for the answer
[1 2]
6
[34] [58]
7
(A + B)² ‡ (A + B).)
(Hint: to show that H is not closed under addition, it is sufficient to find two idempotent matrices A and B such that
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,
(Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number and an idempotent matrix A such that
5 6
(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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F525e4233-5305-4a6a-9e03-c9b07112469a%2F0379f0a1-8ab8-4af4-a7c2-3a7897fbff1d%2Fa1dq8ec_processed.png&w=3840&q=75)
Transcribed Image Text: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], [3,4]], [[5,6],[7,8]] for the answer
[1 2]
6
[34] [58]
7
(A + B)² ‡ (A + B).)
(Hint: to show that H is not closed under addition, it is sufficient to find two idempotent matrices A and B such that
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,
(Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number and an idempotent matrix A such that
5 6
(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
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