The trace of a square n x n matrix A = (aii) is the sum a11+ a22 +...+ ann of the entries on its main diagonal. 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 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 and syntax such as [[1,2],[3,4]], [[5,6],[7,8]] for the (Hint: to show that H is not closed answer 3 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 answer 2, (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 = (aii) is the sum a11+ a22 +...+ ann of the entries on its main diagonal. 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 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 and syntax such as [[1,2],[3,4]], [[5,6],[7,8]] for the (Hint: to show that H is not closed answer 3 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 answer 2, (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
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Chapter 4.1 Question 3
![The trace of a square n x n matrix A =
(aij) is the sum a11 + a22 + · · .+ ann of the entries on its main diagonal.
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
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 and syntax such as [[1,2], [3,4]], [[5,6],[7,8]] for the answer
2
[5
E l: (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
[3
in H, using a comma separated list and syntax such as 2, [[3,4],[5,6]] for the answer 2,
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 r A has nonzero trace.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F57076f46-82b7-4801-86c4-7c157fde2525%2F809f733a-be9e-42a8-b2a7-dc196f76a115%2Fmi32903_processed.png&w=3840&q=75)
Transcribed Image Text:The trace of a square n x n matrix A =
(aij) is the sum a11 + a22 + · · .+ ann of the entries on its main diagonal.
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
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 and syntax such as [[1,2], [3,4]], [[5,6],[7,8]] for the answer
2
[5
E l: (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
[3
in H, using a comma separated list and syntax such as 2, [[3,4],[5,6]] for the answer 2,
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 r A has nonzero trace.)
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