The use of the technique of elimination in solving a system of m linear equations in n variables may be mirrored by using row operations to maneuver the m \times (n + 1) matrix of the system into row echelon form.
One may gain notational compactness in solving such a system of equations by maneuvering its matrix to row echelon form and then applying common sense to solve the resulting system of equations.
The exercises below pertain to the following system of equations:
| w + x - 2 y - z = s |
| 3w - 5 x - 4 y + 3 z = t |
| 2w - 2 x - 3 y + 1 z = u |
| 4w - 8 x - 5 y + 5 z = v |
Find a non-zero point (s, t, u, v) for which the system has no solution.
Find a non-zero point (s, t, u, v) for which the system has at least one solution.
Find linear equations in s, t, u, v that characterize the set of all points (s, t, u, v) for which the system has at least one solution.
Let f and g be point-valued functions of points that are defined by
f(u, v) = (2 u - 3 v, u + v) and g(x, y) = (3 x - y, -x + 2 y) .
Write matrices corresponding to f and g.
Compute the function h that is defined by h(u, v) = g(f(u, v)).
Find a matrix for h.