Exercise 24

Using the idea of a vector space we can easily reprove that the solution set of a homogeneous linear system has either one element or infinitely many elements. Assume that $\vec{v}\in V$ is not $\vec{0}$.

• Prove that $r\cdot\vec{v}=\vec{0}$ if and only if $r=0$.

• Prove that $r_{1}\cdot\vec{v}=r_{2}\cdot\vec{v}$ if and only if $r_{1}=r_{2}$.

• Prove that any nontrivial vector space is infinite.

• Use the fact that a nonempty solution set of a homogeneous linear system is a vector space to draw the conclusion.