The linear system of equations is given: \begin{align} a_{11}x_1+\dots& +a_{1n}x_n=b_1\\ &\vdots\\ a_{m1}x_1+\dots&+a_{mn}x_n=b_m \end{align} Show that the set of the given linear equation system solutions form a sub-vector space of $\mathbb{R^n}$ exactly when $b_i = 0, \;\forall1\leq i \leq m$
"$\leftarrow$" Let $U$ be the solution set of the system of equations and $\forall b_i=0$. Then $0\in U\neq\emptyset$ and it immediately applies that if $u,v\in U: u+v=0+0\iff0\in U$ and $\lambda\in\mathbb{K},\;u\in U: \lambda u = 0\cdot u = 0\in U$ and therefore $U$ is a sub-vector space of $\mathbb{R}^n$
"$\rightarrow$" How to go on? Should I assume, that $\forall b_i\neq0$ and try to show, that the axioms of subspaces only hold for $b_i=0$?
Any vector subspace of $\mathbb R^{n}$ contains the zero vector $(0,0,...,0)$. So the equation must hold when each $x_i$ is $0$ which gives $b_i=0$ for all $i$.