The trace of a square n × n matrix A = (ażj) 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 [1 2] [5 " (Hint: to show that H is not closed under addition, it is sufficient to find two trace zero matrices 3 4 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 [3 2, [³1] . (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r 5 and a trace zero matrix A such that rA has 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 × n matrix A = (ażj) 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 [1 2] [5 " (Hint: to show that H is not closed under addition, it is sufficient to find two trace zero matrices 3 4 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 [3 2, [³1] . (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r 5 and a trace zero matrix A such that rA has 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 × n matrix A = (ażj) 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
[1 2] [5
"
(Hint: to show that H is not closed under addition, it is sufficient to find two trace zero matrices
3 4
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
[3
2,
[³1]
. (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r
5
and a trace zero matrix A such that rA has 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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F05295cc1-71f0-4e3b-b6f8-2c1731c6d617%2Fa20e04e4-0b92-440c-b7b5-bb1f5651d82f%2Feys2smi_processed.png&w=3840&q=75)
Transcribed Image Text:The trace of a square n × n matrix A = (ażj) 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
[1 2] [5
"
(Hint: to show that H is not closed under addition, it is sufficient to find two trace zero matrices
3 4
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
[3
2,
[³1]
. (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r
5
and a trace zero matrix A such that rA has 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
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