Exercise 11

Prove that if $\vec{s}$ and $\vec{t}$ satisfy a homogeneous system then so do these vectors.

• (a)  $\vec{s}+\vec{t}$     (b)  $3\vec{s}$     (c)  $k\vec{s}+m\vec{t}$ for $k,m\in\mathbb{R}$

What’s wrong with this argument: “These three show that if a homogeneous system has one solution then it has many solutions — any multiple of a solution is another solution, and any sum of solutions is a solution also — so there are no homogeneous systems with exactly one solution.”?