For each of the following linear systems, (i) verify...

For each of the following linear systems, (i) verify compatibility using the Fredholm alternative, (ii) find the general solution, and (iii) find the solution of minimum Euclidean norm: $$ \text { (a) } \begin{aligned} 2 x-4 y & =-6, \\ -x+2 y & =3, \end{aligned} $$ $$ x+3 y+5 z=3 \text {, } $$ (d) $$ \begin{aligned} & -x+4 y+9 z=11, \\ & 2 x+3 y+4 z=0, \end{aligned} $$ (e) (b) $$ 2 x+3 y=-1, $$ (c) $$ \begin{aligned} 6 x-3 y+9 z & =12, \\ 2 x-y+3 z & =4, \\ & x-y+2 z+3 w=5 \end{aligned} $$ $$ 2 x-y+3 z=4, $$ $$ \begin{aligned} x_1-3 x_2+7 x_3 & =-8, \\ 2 x_1+x_2 & =5, \end{aligned} $$ $$ x-y+2 z+3 w=5, $$ (f) $$ \begin{aligned} 3 x-3 y+5 z+7 w & =13, \\ -2 x+2 y+z+4 w & =0 . \end{aligned} $$

For each of the following linear systems, (i) verify compatibility using the Fredholm alternative, (ii) find the general solution, and (iii) find the solution of minimum Euclidean norm:
$$
\text { (a) } \begin{aligned}
2 x-4 y & =-6, \\
-x+2 y & =3,
\end{aligned}
$$
$$
x+3 y+5 z=3 \text {, }
$$
(d)
$$
\begin{aligned}
& -x+4 y+9 z=11, \\
& 2 x+3 y+4 z=0,
\end{aligned}
$$
(e)
(b)
$$
2 x+3 y=-1,
$$
(c)
$$
\begin{aligned}
6 x-3 y+9 z & =12, \\
2 x-y+3 z & =4, \\
& x-y+2 z+3 w=5
\end{aligned}
$$
$$
2 x-y+3 z=4,
$$
$$
\begin{aligned}
x_1-3 x_2+7 x_3 & =-8, \\
2 x_1+x_2 & =5,
\end{aligned}
$$
$$
x-y+2 z+3 w=5,
$$
(f)
$$
\begin{aligned}
3 x-3 y+5 z+7 w & =13, \\
-2 x+2 y+z+4 w & =0 .
\end{aligned}
$$

Key Concepts

Fredholm Alternative

The Fredholm Alternative is a fundamental principle in linear algebra that provides necessary and sufficient conditions for the existence of solutions to a linear system. It states that for a given matrix A and a vector b, either the equation Ax = b has a solution, or the homogeneous adjoint equation A?y = 0 has a nontrivial solution which is orthogonal to b. This concept is used to verify the compatibility (or consistency) of linear systems by checking if b lies in the range of A.

General Solution of a Linear System

Finding the general solution of a linear system involves determining one particular solution to the non-homogeneous equation and adding it to the general solution of the associated homogeneous system. The homogeneous solution is derived from the null space (or kernel) of the matrix, and its dimension is given by the nullity of A. Thus, the complete solution set is expressed as the sum of a particular solution and any vector in the null space.

Minimum Euclidean Norm Solution

The minimum Euclidean norm solution is the unique solution that minimizes the Euclidean (L2) norm among all possible solutions of a consistent linear system. This solution is especially important in underdetermined systems and can be obtained using the Moore-Penrose pseudoinverse. It provides the best approximation in terms of least squares and is often computed when a unique solution is required from infinitely many possibilities.

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