The trace of a square n x n matrix A = (a) 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 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 [32] [58] has nonzero trace.) (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 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 [3 4] 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 5 6 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 square n x n matrix A = (a) 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 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 [32] [58] has nonzero trace.) (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 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 [3 4] 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 5 6 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 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 × 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
6
3] [3]
has nonzero trace.)
34
nonzero trace.)
(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
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
[3]
5 6
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%2F8f21868f-81f5-47ff-b0b8-24fe685287c0%2F24334e16-0573-4dea-bf56-b0e92cbaa8d6%2Fxzwc3yj_processed.png&w=3840&q=75)
Transcribed Image Text: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 × 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
6
3] [3]
has nonzero trace.)
34
nonzero trace.)
(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
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
[3]
5 6
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
Expert Solution
Step 1: H is always non empty
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