Geometric interpretation of homogeneous system of equations

Question

• Why does an homogeneous system of linear equations determine a plane that passes through the origin?
• How can I be sure the the vectors that satisfy $Ax = 0$ lie on that same plane?

Answer 1

The solution set of a homogenous system of linear equations is a set $S$ of vectors $\vec{v}$ whose values $v_j$ satisfy a set of $m$ equations $$a_{1,i}v_1+a_{2,i}v_2+...+a_{n,i}v_n=0 \tag{1}$$ where $1\le i\le m$. We can see that the solution set $S$ of this system of $m$ equations (which you are calling a plane, although that name is only applicable when $m=1$ and we are working in three dimensions) passes through the origin because all equations in the form of $(1)$ are trivially true when $\vec{v}=\vec{0}$, or when $v_j=0$ for all $j$: $$a_{1,i}\cdot 0+a_{2,i}\cdot 0+...+a_{n,i}\cdot 0=0$$

As for your second question, this depends on how exactly you define $A$ and is clearly not true in general.