# Showing that $\mathbf{u}=\mathbf{0}$ for $c\mathbf{u}=\mathbf{0}$, $c$ nonzero

The problem is:

Suppose $c\mathbf{u} = \mathbf{0}$ for some nonzero scalar $c$. Show that $\mathbf{u}=\mathbf{0}$.

Using the axioms for a vector space, I figured I could use $c(d\mathbf{u}) = (cd)\mathbf{u}$ first, and multiplying both sides with the inverse of $c$: \begin{align} c\mathbf{u}=\mathbf{0} \\ c^{-1}(c\mathbf{u})=\mathbf{0} c^{-1} \\ (c^{-1}c)\mathbf{u}=\mathbf{0} \\ 1\mathbf{u} = \mathbf{0} \end{align}

And then use the axiom that $1 \mathbf{u} = \mathbf{u}$ to show that $\mathbf{u} = \mathbf{0}$.

Is this somewhat correct? Is there a better, or more precise way of doing this? Thank you!

-
Yes, it looks perfectly fine. – user9413 Sep 4 '11 at 13:49
@Chandrasekhar: Thanks! – Jodles Sep 4 '11 at 13:50
And to be nitpicky, I don't know if it's acceptable to everyone to write $\mathbf 0 c^{-1}$. The scalar is usually written to the left of the vector. – Srivatsan Sep 4 '11 at 15:05
@Srivatsan: Good to know, thanks! – Jodles Sep 4 '11 at 15:57

Depending on the definition of a vector space, you may need to explain why $c^{-1}0=0$ (which is used on the RHS of the third line of your proof). In many (or most?) textbook definitions of a vector space, that "the scalar multiple of a zero vector is the zero vector itself" is not part of the definition, but a consequence of the axioms.