Question

Let a,b,c,d be constants and let U={[x,y,z] R3: ax+by+cz=0}. Show that U is a subspace of V if and only if d=0. That is, a plane in space is a subspace if and only if it passes though the origin 0.

Answer 1

\[\left\{ \left(\begin{matrix}x \\ y\\z\ \end {matrix}\right)\epsilon \mathbb{R} ^3 : ax+by+cz=d \right\}\]