= Solution
Expand the right-hand side in the <orthonormal basis> from part (b). Its coefficients are
$$
\mathbf b\cdot\mathbf e_0=0,\qquad\mathbf b\cdot\mathbf e_4=2\sqrt2,\qquad\mathbf b\cdot\mathbf e_6=\sqrt3.
$$
Thus $\mathbf b=2\sqrt2\mathbf e_4+\sqrt3\mathbf e_6$. If $\mathbf x=c_0\mathbf e_0+c_4\mathbf e_4+c_6\mathbf e_6$, then <matrix multiplication> gives $A\mathbf x=4c_4\mathbf e_4+6c_6\mathbf e_6$. Matching coefficients fixes $c_4=\sqrt2/2$ and $c_6=\sqrt3/6$, while $c_0$ is arbitrary. Consequently all solutions are
$$
\boxed{\mathbf x=\begin{pmatrix}2/3\\-1/3\\1/6\end{pmatrix}+t\begin{pmatrix}1\\1\\-2\end{pmatrix},\qquad t\in\mathbb R.}
$$
Geometrically, the solution set is an <affine line in a vector space> parallel to the <kernel> of $A$. The zero <eigenvalue> makes the component in that direction invisible to $A$; the other two components are uniquely determined. The displayed particular solution is perpendicular to the <kernel>, so it is also the point of this <line> closest to the origin.
Back to article page
