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 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 4 7 closed under addition, it is sufficient to find two trace zero matrices A and B such that A + B has nonzero trace.) [[1,0],[0,0]], [[0,1],[2,1]] 6]. 8 (Hint: to show that H is not 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.) 2, [[0,0],[1,1]] 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 not 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 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 4 7 closed under addition, it is sufficient to find two trace zero matrices A and B such that A + B has nonzero trace.) [[1,0],[0,0]], [[0,1],[2,1]] 6]. 8 (Hint: to show that H is not 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.) 2, [[0,0],[1,1]] 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 not 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
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Hi. I attempted this question and got it wrong. Please can you help me with the solutions
![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
6
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
closed under addition, it is sufficient to find two trace zero matrices A and B such that A + B has nonzero trace.)
[[1,0], [0,0]], [[0,1],[2,1]]
3].[53].
(Hint: to show that H is not
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 answer 2,
. (Hint: to show that H is
2 [ ³3 ].
5
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.)
2,[[0,0],[1,1]]
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 not a subspace of V](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd29ed1ca-eabb-4fe1-ba2f-df7a36f73903%2Fb095ab59-2655-4978-bc8c-a27fd4540fd7%2Frs32xz_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 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
6
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
closed under addition, it is sufficient to find two trace zero matrices A and B such that A + B has nonzero trace.)
[[1,0], [0,0]], [[0,1],[2,1]]
3].[53].
(Hint: to show that H is not
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 answer 2,
. (Hint: to show that H is
2 [ ³3 ].
5
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.)
2,[[0,0],[1,1]]
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 not a subspace of V
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