Question

The trace of a square n×nn×n matrix A=(aij)A=(aij) is the sum a11+a22+⋯+anna11+a22+⋯+ann of the entries on its main diagonal.

Let VV be the vector space of all 2×22×2 matrices with real entries. Let HH be the set of all 2×22×2 matrices with real entries that have trace 11. Is HH a subspace of the vector space VV?

1. Does HH contain the zero vector of VV?
   choose H contains the zero vector of V H does not contain the zero vector of V
   
2. Is HH closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in HH whose sum is not in HH, using a comma separated list and syntax such as [[1,2],[3,4]], [[5,6],[7,8]][[1,2],[3,4]], [[5,6],[7,8]] for the answer [1324],[5768][1234],[5678]. (Hint: to show that HH is not closed under addition, it is sufficient to find two trace one matrices AA and BB such that A+BA+B has trace not equal to one.)
   
   
3. Is HH closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in RR and a matrix in HH whose product is not in HH, using a comma separated list and syntax such as 2, [[3,4],[5,6]]2, [[3,4],[5,6]] for the answer 2,[3546]2,[3456]. (Hint: to show that HH is not closed under scalar multiplication, it is sufficient to find a real number rr and a trace one matrix AA such that rArA has trace not equal to one.)
   
   
4. Is HH a subspace of the vector space VV? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3.

Answer 1

Let. V be a vector space. A subset W of V is a sub space of V iff zero vector of V belongs to W, W is closed under addition and W is closed under scalar multiplication.