## Affine hull

Show that for any two solutions $$\mathbf x=(x_1,x_2,…,x_n)^T$$ and $$\mathbf x'=(x_1',x_2',…,x_n')^T$$ of a given system we get also a solution $$\alpha\mathbf x+(1-\alpha)\mathbf x'=(\alpha x_1+ (1-\alpha) x_1',\alpha x_2+ (1-\alpha) x_2',…,\alpha x_n+ (1-\alpha) x_n')^T$$ for an arbitrary real $$\alpha$$.
Generalize your argument also for more solutions $$\mathbf x,\mathbf x',…,\mathbf x^{(k)}$$.
Any equation of the system $$a_{i1}x_1 + … + a_{in}x_n = b_i$$ is satisfied also by the new solution
$$(\alpha x_1+ (1-\alpha) x_1',\alpha x_2+ (1-\alpha) x_2',…,\alpha x_n+ (1-\alpha) x_n')^T$$, because
$$a_{i_1}(\alpha x_1+ (1-\alpha) x_1')+…+a_{in}(\alpha x_n+ (1-\alpha) x_n')=$$
$$= \alpha (a_{i1}x_1 + … + a_{in}x_n) + (1-\alpha)(a_{i1}x_1' + … + a_{in}x_n')= \alpha b_i + (1-\alpha) b_i = b_i$$
In general, it is possible to take any affine combination of $$\mathbf x,\mathbf x',…,\mathbf x^{(k)}$$, i.e. $$\alpha_0 \mathbf x + \alpha_1 \mathbf x' + … + \alpha_k \mathbf x^{(k)}$$, where the coefficients satisfy $$\alpha_0 + … + \alpha_k = 1$$.