Question

Check whether or not the set of all symmetric matrix in Mn(R) form a real vector space with respect to the usual addition and scalar multiplication for Mn(R)

Answer 1

By definition, a real vector space is a non-empty set (it is true for $M_n \lparen \Reals \rparen$ ) equipped with operations of addition and multiplication with a real number which satisfy the following axioms:

1. $\forall A, B \in M_n \lparen \Reals \rparen A + B = B + A$ ( it is true for arbitrary matrices)
2. $\forall A, B, C \in M_n \lparen \Reals \rparen (A + B) + C = A + (B + C)$ (it is true for arbitrary matrices)
3. $\exists \theta \in M_n (\Reals) : \forall A \in M_n (\Reals) A + \theta = \theta + A = A$ (it is true because the zero matrix is a symmetric matrix)
4. $\forall A \in M_n (\Reals) \exists (-A) \in M_n (\Reals) : \\ A + (-A) = (-A) + A = \theta$ (it follows from the matrix transposition properties)
5. $\exists I \in M_n (\Reals) : \forall A \in M_n (R) I \cdot A = A$ (it is true because the identical matrix is a symmetric matrix)
6. $\forall \alpha , \beta \in \Reals , \forall A \in M_n (\Reals) (\alpha \cdot \beta) \cdot A = \alpha \cdot (\beta \cdot A)$ (it follows from the matrix multiplication with scalar properies)
7. $\forall \alpha \in \Reals , \forall A, B \in M_n (\Reals) \alpha \cdot (A + B) = \\ = \alpha \cdot A + \alpha \cdot B$ (it follows from the matrix multiplication with scalar properties)
8. $\forall \alpha , \beta \in \Reals , \forall A \in M_n (\Reals) (\alpha + \beta) \cdot A = \\ = \alpha \cdot A + \beta \cdot A$ (it follows from the matrix multiplication with scalar properties)

Thus, $M_n (\Reals)$ is a real vector space.