The trace of a squaren x n matrix A = (aj) is the sum a₁1 + a22+...+ann of the entries on its main diagonal. 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 matrices with real entries that have trace 0. 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 (Hint: to show that H is not closed under addition, it is sufficient to find two trace zero matrices A and B such that A + has nonzero trace.) [369] 2². [³4]. 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 a trace zero matrix A such that rA has nonzero trace.) 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
The trace of a squaren x n matrix A = (aj) is the sum a₁1 + a22+...+ann of the entries on its main diagonal. 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 matrices with real entries that have trace 0. 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 (Hint: to show that H is not closed under addition, it is sufficient to find two trace zero matrices A and B such that A + has nonzero trace.) [369] 2². [³4]. 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 a trace zero matrix A such that rA has nonzero trace.) 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
Related questions
Question
![The trace of a square î × ʼn matrix A = (aij) is the sum a11 + a22 +・・・ + ann of the entries on its main diagonal.
V?
Let V be the vector space of all 2 × 2 matrices with real entries. Let H be the set of all 2 × 2 matrices with real entries that have trace 0. Is H a subspace of the vector space
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
(Hint: to show that H is not closed under addition, it is sufficient to find two trace zero matrices A and B such that A + B has nonzero trace.)
[33]
4
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
5
6
7 8
[33]
5
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 a trace zero matrix A such that rA has nonzero trace.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ff93d80-f747-4f15-803d-56788ff3d116%2F080ce920-a826-4e53-9d04-cea2ce83a84a%2Fk7k2wzb_processed.png&w=3840&q=75)
Transcribed Image Text:The trace of a square î × ʼn matrix A = (aij) is the sum a11 + a22 +・・・ + ann of the entries on its main diagonal.
V?
Let V be the vector space of all 2 × 2 matrices with real entries. Let H be the set of all 2 × 2 matrices with real entries that have trace 0. Is H a subspace of the vector space
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
(Hint: to show that H is not closed under addition, it is sufficient to find two trace zero matrices A and B such that A + B has nonzero trace.)
[33]
4
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
5
6
7 8
[33]
5
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 a trace zero matrix A such that rA has nonzero trace.)
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