According to the addition of matrix and the multiplication of number and matrix, does the set of square matrices in the following real number field constitute a linear space in the real number field (1) (2) the set of all symmetric matrices of order n; (3) a is the set of known matrices of order n satisfying AX = 0

According to the addition of matrix and the multiplication of number and matrix, does the set of square matrices in the following real number field constitute a linear space in the real number field (1) (2) the set of all symmetric matrices of order n; (3) a is the set of known matrices of order n satisfying AX = 0

(1) Yes
(2) Yes
(3) Yes
Because the set of square matrices of the same order is a linear space
So we only need to prove that the addition and multiplication of matrix are closed
For example, (2) the sum of symmetric matrix is still symmetric matrix; K times of symmetric matrix is still symmetric matrix