Question

Let a be an mn matrix, and let u and v be vectors in irn with the property that au = 0 and av = 0. explain why a(u + v) must be the zero vector. then explain why a(cu + dv) = 0 for each pair of scalars c and
d.

Answer 1

$\mathbf{Au}=\mathbf0$
$\mathbf{Av}=\mathbf0$

$\implies\mathbf{Au}+\mathbf{Av}=\mathbf0$
$\implies\mathbf A(\mathbf u+\mathbf v)=\mathbf0$

Multiplying by scalars $c,d$ won't change this fact.