{{1, 1, s}, {1, s, 1}, {s, 1, 1}}

for what value of s is the solution space of the origin only, line through origin, plane through origin, or all of R^3

for origin only the s=0 give the result x=0, y=0, z=0

for plane only the s=1 give the result x=-y-z

for a line s=? should give the result x=y, z=0. How can do this without having 2 different value of s?

for all of R^3 i have no idea what to do. Considering it has to be linearly independent then all s value should be the same?