At least one of the answers above is NOT correct. 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 Ha 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 [12] [56] and syntax such as [[1,2], [3,4]], [[5,6],[7,8]] for the answer . (Hint: to show that H is not closed under 3 4 7 [13 2 addition, it is sufficient to find two idempotent matrices A and B such that (A + B)² ‡ (A + B).) [[1,0], [0,0]], [[0,0], [0,1]] 6 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, [3 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 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
At least one of the answers above is NOT correct. 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 Ha 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 [12] [56] and syntax such as [[1,2], [3,4]], [[5,6],[7,8]] for the answer . (Hint: to show that H is not closed under 3 4 7 [13 2 addition, it is sufficient to find two idempotent matrices A and B such that (A + B)² ‡ (A + B).) [[1,0], [0,0]], [[0,0], [0,1]] 6 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, [3 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 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
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter2: Matrices
Section2.CR: Review Exercises
Problem 37CR
Related questions
Question
![At least one of the answers above is NOT correct.
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 Ha
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
[12] [56]
and syntax such as [[1,2], [3,4]], [[5,6],[7,8]] for the answer
. (Hint: to show that H is not closed under
3 4 7
[13
2
addition, it is sufficient to find two idempotent matrices A and B such that (A + B)² ‡ (A + B).)
[[1,0], [0,0]], [[0,0], [0,1]]
6
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,
[3
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 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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7f90e306-fe27-48c9-8d70-31d0c29a7db1%2Fc38538c3-d273-4ebb-bd8c-29cbe4f786c3%2Fiqf5kq8g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:At least one of the answers above is NOT correct.
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 Ha
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
[12] [56]
and syntax such as [[1,2], [3,4]], [[5,6],[7,8]] for the answer
. (Hint: to show that H is not closed under
3 4 7
[13
2
addition, it is sufficient to find two idempotent matrices A and B such that (A + B)² ‡ (A + B).)
[[1,0], [0,0]], [[0,0], [0,1]]
6
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,
[3
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 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
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