Let G be the 4 \times 4 matrix
| , |
and let f be the linear map (or function) from R^{4} to R^{4} defined by the formula
| y = f(x) = G x . |
Solve each of the following systems of 4 linear equations in 4 unknowns x_{1}, x_{2}, x_{3} and x_{4}.
f(x) = (0, 0, 0, 0).
f(x) = (1, -1, 1, 3) with x_{3} = 0.
f(x) = (1, -1, 1, 4) with x_{3} = 0.
f(x) = (1, -1, 1, 4) with x_{3} = x_{4} = 0.
f(x) = (3, -1, 2, 1) with x_{3} = 0.
f(x) = (3, -1, 7, 10) with x_{3} = 0.
Answer the following questions:
What is the kernel of f ?
Find equations that characterize the image of f.
For each part of the first preceding problem if there are solutions find a solution s and a minimal set u, v, ... of vectors such that the most general solution of the system is the sum of s and an arbitrary linear combination of the vectors u, v, ....