Show in general ie prove whether or not the set of symmetric

Show in general (i.e. prove) whether or not the set of symmetric matrices (A = A^T) is a subspace of Ropf^3 times 3 (i.e. a subspace of real matrices size 3 times 3: M_3 times 3 (Ropf)).

Solution

1. Let, A, B be two symmetric matrices

(A+B)^T=A^T+B^T=A+B

Hence, A+B is also symmetric

2. Let, A be symmetric and c be a real number

(cA)^T=cA^T=cA

Hence, cA is also symmetric

HEnce, set of symmetirc matrices is a subspace