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The trace of a square ʼn x n matrix A = (a¿j) is the sum a11 + a22 + entries on its main diagonal. ... 1. Is H nonempty? choose +ann of the 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? 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 5 6 [[1,2], [3,4]], [[5,6], [7,8]] for the answer (Hint: to show that His 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 in H, using a comma separated list and syntax such 3 4 as 2, [[3,4], [5,6]] for the answer 2, . (Hint: to show that H is not closed 6 under scalar multiplication, it is sufficient to find a real number 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 ʼn x n matrix A = (a¿j) is the sum a11 + a22 + entries on its main diagonal. ... 1. Is H nonempty? choose +ann of the 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? 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 5 6 [[1,2], [3,4]], [[5,6], [7,8]] for the answer (Hint: to show that His 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 in H, using a comma separated list and syntax such 3 4 as 2, [[3,4], [5,6]] for the answer 2, . (Hint: to show that H is not closed 6 under scalar multiplication, it is sufficient to find a real number 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

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.4: Orthogonal Diagonalization Of Symmetric Matrices
Problem 26EQ
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The trace of a square ʼn x n matrix A = (a¿j) is the sum a11 +a22 + entries on its main diagonal. ... 1. Is H nonempty? choose +ann of the 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? 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 5 6 [[1,2], [3,4]], [[5,6], [7,8]] for the answer (Hint: to show that His 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 in H, using a comma separated list and syntax such 3 4 as 2, [[3,4], [5,6]] for the answer 2, . (Hint: to show that H is not closed 6 under scalar multiplication, it is sufficient to find a real number 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
Transcribed Image Text:The trace of a square ʼn x n matrix A = (a¿j) is the sum a11 +a22 + entries on its main diagonal. ... 1. Is H nonempty? choose +ann of the 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? 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 5 6 [[1,2], [3,4]], [[5,6], [7,8]] for the answer (Hint: to show that His 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 in H, using a comma separated list and syntax such 3 4 as 2, [[3,4], [5,6]] for the answer 2, . (Hint: to show that H is not closed 6 under scalar multiplication, it is sufficient to find a real number 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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