The trace of a square n 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 1 5 6 3 (Hint: to show that H is not closed under addition, it is sufficient to find two trace zero 4 matrices A and B such that A + B has nonzero trace.) 8 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 ·[31]. (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a 6 real number and a trace zero matrix A such that rA has nonzero trace.) answer 2, 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 square n 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 1 5 6 3 (Hint: to show that H is not closed under addition, it is sufficient to find two trace zero 4 matrices A and B such that A + B has nonzero trace.) 8 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 ·[31]. (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a 6 real number and a trace zero matrix A such that rA has nonzero trace.) answer 2, 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
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section: Chapter Questions
Problem 17RQ
Related questions
Question
![The trace of a square n x n matrix A = (a¡j) is the sum a₁ + 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
(3 3.55 3.
(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.)
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
(Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a
answer 2,
5
real number r 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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F49f6f82d-aa0c-408d-8119-6952b6847c51%2Fd0763538-dc47-4825-970c-c003b8277fe2%2Fke2p87c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The trace of a square n x n matrix A = (a¡j) is the sum a₁ + 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
(3 3.55 3.
(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.)
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
(Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a
answer 2,
5
real number r 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
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