The trace of a square n x n matrix A = (a;j) 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? 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] [5 6] 3 4 (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.) 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, 3 4 - (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r and a trace zero matrix A such that rA has nonzero trace.)
The trace of a square n x n matrix A = (a;j) 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? 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] [5 6] 3 4 (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.) 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, 3 4 - (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r and a trace zero matrix A such that rA has nonzero trace.)
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 n x n matrix A = (a;j) 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?
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] [5 6]
3 4
(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.)
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,
3 4
- (Hint: to show that H is not closed
under scalar multiplication, it is sufficient to find a real number r 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%2Fd941c82a-9f8d-4007-9395-937d5f89c0b4%2F136bd632-2eff-4e21-9b9d-c656d0826f5f%2F0ztc6eh_processed.png&w=3840&q=75)
Transcribed Image Text:The trace of a square n x n matrix A = (a;j) 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?
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] [5 6]
3 4
(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.)
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,
3 4
- (Hint: to show that H is not closed
under scalar multiplication, it is sufficient to find a real number r and a trace zero matrix A such that rA has nonzero trace.)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
