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...

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Chapter 4.1 Question 3

Extracted text: 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 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, : 1. (Hint: to show that H is not closed under scalar 5 6 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. H is a subspace of V