Verify if symmetric matrices form a subspace

Question

I need to verify if the symmetric matrices form a subspace. But I don't know how to represent a general symmetric matrix. I know that the matrix $A$ is symmetric if $A = A^t$ but I can't write a general one and add with another to verify the conditions for a subspace.

Answer 1

A symmetric matrix is one such that $A^t = A$. because the adjoint is a linear map, you know that $(A+B)^t = (A^t +B^t)$.

If you want to be more elementary, we can represent a generic nxn symmetric matrix as a matrix $(a_{i,j})$ such that $a_{i,j} = a_{j,i}$, and same for a matrix $(b_{i,j})$. Because matrix addition is component-wise, the matrix A+B has slot i,j as $a_{i,j}+b_{i,j}$ and slot j, i as $a_{j,i}+b_{j,i}$ which are equal, so A+B is symmetric.