The trace of a square n x n matrix A = (a;;) is the sum a11 + a2+ diagonal. + ann of the entries on its main ... 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. Does H contain the zero vector of V? H contains the zero vector of 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 [[1,2], [3,4]], [[5,6],[7,8]] for the [1 2] [5 6 : 1E 1:(Hint: to show that H is not closed under addition, it is sufficient to find two 3 4 answer 7 8 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 as [3 4] 2, [[3,4], [5,6]] for the answer 2, (Hint: to show that H is not closed under scalar 5 6

The trace of a square n x n matrix A = (a;;) is the sum a11 + a2+ diagonal. + ann of the entries on its main ... 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. Does H contain the zero vector of V? H contains the zero vector of 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 [[1,2], [3,4]], [[5,6],[7,8]] for the [1 2] [5 6 : 1E 1:(Hint: to show that H is not closed under addition, it is sufficient to find two 3 4 answer 7 8 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 as [3 4] 2, [[3,4], [5,6]] for the answer 2, (Hint: to show that H is not closed under scalar 5 6

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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