Affine hull

Task number: 2385

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 \(\mathbf x,\mathbf x',…,\mathbf x^{(k)}\).

  • Resolution

    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 \(\alpha_0 + … + \alpha_k = 1\).

Difficulty level: Easy task (using definitions and simple reasoning)
Proving or derivation task
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