- #1
member 587159
Homework Statement
Prove that in any vector space V, we have:
##\alpha \overrightarrow a = \overrightarrow 0 \Rightarrow \alpha = 0 \lor \overrightarrow a = \overrightarrow 0##
Homework Equations
I already proved:
##\alpha \overrightarrow 0 = \overrightarrow 0##
##0 \overrightarrow a = \overrightarrow 0##
The Attempt at a Solution
[/B]
Suppose ##\alpha \neq 0##
Then: ##\alpha \overrightarrow a = \overrightarrow 0##
##\Rightarrow \alpha^{-1} (\alpha \overrightarrow a) = \alpha^{-1} \overrightarrow 0##
##\Rightarrow (\alpha^{-1} \alpha) \overrightarrow a = \overrightarrow 0##
##\Rightarrow 1 \overrightarrow a = \overrightarrow 0##
##\Rightarrow \overrightarrow a = \overrightarrow 0##
The problem is. I don't know how to show that ##\alpha \overrightarrow a = \overrightarrow 0## can imply ##\alpha = 0## I can't suppose ##\alpha = 0##, because I need to prove that?
Maybe something like this?
##\alpha \overrightarrow a = \overrightarrow 0##
But ##0 \overrightarrow a = \overrightarrow 0##
Thus: ##\alpha \overrightarrow a = 0 \overrightarrow a ##
Comparing the two, we obtain ##\alpha = 0##
Thanks in advance.