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The trace of a square n x n matrix A = (a,j) is the sum a11+ a22 + entries on its main diagonal. +ann of the Let V be the vector space of all 2 × 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. Does H contain the zero vector of V? 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 2 5 6 (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 has nonzero trace.) [34] 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, 5 (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.) 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

The trace of a square n x n matrix A = (a,j) is the sum a11+ a22 + entries on its main diagonal. +ann of the Let V be the vector space of all 2 × 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. Does H contain the zero vector of V? 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 2 5 6 (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 has nonzero trace.) [34] 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, 5 (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.) 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

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.6: Rank Of A Matrix And Systems Of Linear Equations
Problem 69E: Consider an mn matrix A and an np matrix B. Show that the row vectors of AB are in the row space of...
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The trace of a square n x n matrix A = (a;;) is the sum a11 + a22 + entries on its main diagonal. +ann of the 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. Does H contain the zero vector of V? 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 [12] [56] such as [[1,2], [3,4]], [[5,6], [7,8]] for the answer 78 (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 has nonzero trace.) 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter [34] 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, 5 6 (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.) 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 n x n matrix A = (a;;) is the sum a11 + a22 + entries on its main diagonal. +ann of the 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. Does H contain the zero vector of V? 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 [12] [56] such as [[1,2], [3,4]], [[5,6], [7,8]] for the answer 78 (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 has nonzero trace.) 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter [34] 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, 5 6 (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.) 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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