一題空間向量的問題

(a) Let M and N be the elements of S. Then
(1,1,1)(xM+yN)
=x(1,1,1)M+y(1,1,1)N
=(0,0,0)
Also, the zero matrix O is an element of S. So, S is a vector space.

(b) Consider the structure of S
a b c
b e f
c f i

By (1,1,1)S=(0,0,0). we have a+b+c=0, b+e+f=0, c+f+i=0

That is c=-a-b,e=-b-f,i=a+b-f. This implies that row three can be obtained by row one plus row two. On the other hand, it is obvious that row one and two are independent. So, the dimension of S should be 2.