Find the general solution of the homogeneous equation.

Question

Consider the system of equations
\[
\begin{aligned}
&x+y-z=a \\
&x-y+2 z=b \\
&3 x+y=c
\end{aligned}
\]
a) Find the general solution of the homogeneous equation.
b) If \(a=1, b=2\), and \(c=4\), then a particular solution of the inhomogeneous equations is \(x=1, y=1, z=1\). Find the most general solution of these inhomogeneous equations.
c) If \(a=1, b=2\), and \(c=3\), show these equations have no solution.
d) If \(a=0, b=0, c=1\), show the equations have no solution. [Note: \(\left(\begin{array}{l}0 \\ 0 \\ 1\end{array}\right)=\) \(\left.\left(\begin{array}{l}1 \\ 2 \\ 4\end{array}\right)-\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)\right]\).
e) Let \(A=\left(\begin{array}{rrr}1 & 1 & -1 \\ 1 & -1 & 2 \\ 3 & 1 & 0\end{array}\right) .\) Find a basis for \(\operatorname{ker}(A)\) and \(\operatorname{image}(A)\).