Explain why each of the following subsets is or is not a vec

Explain why each of the following subsets is, or is not, a vector subspace. (No credit will be given without explanation/proof)

(a) The set of n x n symmetric matricies; a matrix A is symmetric if a_ij= a_ji for all i and j.

Solution

First of all, Lets understand what is subspace :
If a W is a vector space itself (which means that it is closed under operations of addition and scalar multiplication), with the same vector space operations as V has, then W is a subspace of V.
Now its a subset of n x n symmetric matrices , these matrices can be added to each other or they can be multiplied with each other.
For example, A is nxn matrix and B is also a nxn matrix then these can be added as C=A+B or they can be multiplied as D=AxB where order of D is nxn.