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 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? H is nonempty 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 [36] trace zero matrices A and B such that A + B has nonzero trace.) CLOSED 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 3 4 6 is sufficient to find a real number r and a trace zero matrix A such that rA has nonzero trace.) CLOSED 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. H is a subspace of V
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 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? H is nonempty 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 [36] trace zero matrices A and B such that A + B has nonzero trace.) CLOSED 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 3 4 6 is sufficient to find a real number r and a trace zero matrix A such that rA has nonzero trace.) CLOSED 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. H is a subspace of V
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 = (aij) is the sum a11 + a22 +
1. Is H nonempty?
+ Ann
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 O. Is H a subspace of
the vector space V?
H is nonempty
of the entries on its main diagonal.
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
1 2 5 6
[ 4.6 3].
3
7
8
trace zero matrices A and B such that A + B has nonzero trace.)
CLOSED
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
4
[³3 }]
5
6
is sufficient to find a real number r and a trace zero matrix A such that rA has nonzero trace.)
CLOSED
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.
H is a subspace of V](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3688a312-1b6e-40d4-973f-537807921925%2F9c353ed8-73f7-4992-863b-39e76746375a%2Flfdw7a2_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 +
1. Is H nonempty?
+ Ann
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 O. Is H a subspace of
the vector space V?
H is nonempty
of the entries on its main diagonal.
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
1 2 5 6
[ 4.6 3].
3
7
8
trace zero matrices A and B such that A + B has nonzero trace.)
CLOSED
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
4
[³3 }]
5
6
is sufficient to find a real number r and a trace zero matrix A such that rA has nonzero trace.)
CLOSED
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.
H is a subspace of V
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