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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 $\operatorname{image}(A)$.
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