The set of all symmetric matrices: $\{A \in V : A = A^T\}$

The set of all matrices with determinant zero: $\{A \in V : \det(A) = 0\}$

The set of all diagonal matrices: $\{A \in V : A_{12} = A_{21} = 0\}$

The set of all matrices with trace zero: $\{A \in V : \text{tr}(A) = 0\}$