Page 357 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Consider the linear systems
(a)
(b)
(c)
(d)

Each of these systems has three unknowns, so the solutions form subspaces of . Geometrically, this means that each
solution space must be the origin only, a line through the origin, a plane through the origin, or all of . We shall now verify
that this is so (leaving it to the reader to solve the systems).

Solution

(a) The solutions are

from which it follows that as a normal vector.
This is the equation of the plane through the origin with
(b) The solutions are

which are parametric equations for the line through the origin parallel to the vector
.

(c) The solution is , , , so the solution space is the origin only—that is, .

(d) The solutions are

where r, s, and t have arbitrary values, so the solution space is all of .