1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Vector space property

  1. Oct 3, 2016 #1

    Math_QED

    User Avatar
    Homework Helper

    1. The problem statement, all variables and given/known data

    Prove that in any vector space V, we have:

    ##\alpha \overrightarrow a = \overrightarrow 0 \Rightarrow \alpha = 0 \lor \overrightarrow a = \overrightarrow 0##

    2. Relevant equations

    I already proved:

    ##\alpha \overrightarrow 0 = \overrightarrow 0##
    ##0 \overrightarrow a = \overrightarrow 0##

    3. The attempt at a solution

    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.
     
  2. jcsd
  3. Oct 3, 2016 #2

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member
    2017 Award

    I suggest you focus on the implications of assuming that ##\vec a \neq 0##.
     
  4. Oct 3, 2016 #3

    fresh_42

    User Avatar
    2017 Award

    Staff: Mentor

    You can distinguish cases. Either ##\alpha = 0## or ##\alpha \neq 0##. One of the two has to happen.
     
  5. Oct 3, 2016 #4

    Math_QED

    User Avatar
    Homework Helper

    But can I assume ##\alpha = 0##? Then it follows trivially that ##\alpha = 0 \land \alpha \overrightarrow a = \overrightarrow 0 \Rightarrow \alpha = 0##
     
  6. Oct 3, 2016 #5

    fresh_42

    User Avatar
    2017 Award

    Staff: Mentor

    Why not?
    $$A = A \wedge \text{ true } = A \wedge (B \vee \lnot B) = (A \wedge B) \vee (A \wedge \lnot B)$$
    and ##B=(\alpha = 0)## does the job.
     
  7. Oct 3, 2016 #6

    Math_QED

    User Avatar
    Homework Helper

    Nice. Thanks a lot.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted