2
$\begingroup$

If $V$ is a real vector space then we know that cancellation law in addition holds.

Am I right to say that $ax=ay \Rightarrow x=y$ whenever $a\ne 0$ and $x,y$ are vectors?

3 Answers 3

3

To elaborate a little on what alex.jordan said...

The scalars for a vector space lie in a structure called a field, which (if you don't know) implies that every scalar $a \ne 0$ has a multiplicative inverse $a^{-1}$ such that $aa^{-1}=a^{-1}a=1$. So if $ax=ay$ where $a$ is a scalar and $x$,$y$ are vectors, then $ax=ay \Rightarrow a^{-1}ax=a^{-1}ay \Rightarrow 1x=1y \Rightarrow x=y$.

4

If $a\neq 0$, then $x=1x=(\frac{1}{a}a)x=\frac{1}{a}(ax)=\frac{1}{a}(ay)=(\frac{1}{a}a)y=1y=y$.

3

Yes. To prove it, multiply both sides by $a^{-1}$. And use the axiom that $1\cdot\vec{v}=\vec{v}$.