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] answer [33] [58]- (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).) closed 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, [36]. (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).) closed 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
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] answer [33] [58]- (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).) closed 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, [36]. (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).) closed 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
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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Please answer correctly following the same format as the questions. Box the correct answer and I will give a thumbs up if the answer is correct, a thumbs down if answer is incorrect.
Chapter 4.1 Question 6
Please answer the two red boxes
![You have answered 2 out of 4 parts correctly.
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
2]
answer
[158]. (Hint: to show that H is not closed under addition, it is sufficient to find two
3
idempotent matrices A and B such that (A + B)² ‡ (A + B).)
closed
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, [34]. (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).)
closed
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b030905-8b33-4143-9146-4ba73429bad6%2Fd5820120-eeec-4e1a-94a4-8933c187848f%2F9lf790u_processed.png&w=3840&q=75)
Transcribed Image Text:You have answered 2 out of 4 parts correctly.
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
2]
answer
[158]. (Hint: to show that H is not closed under addition, it is sufficient to find two
3
idempotent matrices A and B such that (A + B)² ‡ (A + B).)
closed
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, [34]. (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).)
closed
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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