Exercise-2

LADR

Exercise 1

Suppose \(a \in \mathbf{F}, v \in V\), and \(av = 0\). Prove that \(a = 0\) or \(v = 0\).

Solution

Let us assume that \(a \neq 0\) to begin with:

\[ \begin{aligned} v &= 1v\\ &= \left(a \cdot \cfrac{1}{a} \right) v\\ &= \cfrac{1}{a} (a v)\\ &= \cfrac{1}{a} 0\\ &= 0 \end{aligned} \]

We have shown that if \(a \neq 0\) then \(v = 0\). If \(v \neq 0\) then \(a\) has to be \(0\). This is just the contrapositive of what has been proven.

Footnotes

  1. Exercise-2, Exercises 1B, Page-16, Linear Algebra Done Right, Fourth Edition, Sheldon Axler↩︎