Question

(1 point) The trace of a square n x n matrix A = (aii) is the sum ani + 022 + ... + ann of the entries on its main diagonal. 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 1. Is Ha subspace of the vector space V? 1. Does H contain the zero vector of V? H does not contain 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 1 for the answer [1 2] [5 6 ] Hint syntax such as [[1,2],[3,4]], [[5,6],[7,8]] for the answer . (Hint: to show that H is not closed under addition, it is sufficient to wer [34]'17 8]." find two trace one matrices A and B such that A + B has trace not equal to one.) 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 . [34 comma separated list and syntax such as 2, [[3,4],[5,6]] for the answer 2, 25 6. Hint (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r and a trace one matrix A such that rA has trace not equal to one.) 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 not a subspace of V

Answer 1

subspaces are nonempty and are closed under sums and under scalar multiples.

The detailed solution is attached below.

let v be matrices the vector with real space entries of all 2x2 let H be the real entries set that of 2x2 matrices with have trace I. Lo ou 1. We see that zero vector is to o whose trace is equal to ofl szero element does not belong to H. 2. H is not closed under addition Tool to JCH But [ o + Lo g - 11 o whose | = 6001L001H OJ Lo o trace is 2 1

3. His not closed multiplication under scalar o en But alio 12 o whose Lo Lo od Lool trace is equal to 2 & 1 - 2o & H Loofa 4. By that parts 1-3 it is not we conclude a subspace of v.