Simple Vector Space Proof

Oct 2008
100
9
Let V be a vector space over F, and let a be an element of F and v be an element of V.

Prove:
If av=0, then a=0 or v=0

My attempt:
Assume \(\displaystyle a\ne0\) and \(\displaystyle v\ne0\)
\(\displaystyle v\ne v+v\) since the 0 vector is unique (this is proven in the book)
\(\displaystyle <=> av\ne av+av\) if this were an equality we could multiply both sides by (1/a) and obtain the negation of the last statement
\(\displaystyle => av \ne 0\)
Since we know av=0, the assumption is wrong, and the negation of the assumption is the statement we want to prove.

Am I right? Even if I am, do you know a clearer/better way to do it?

Thanks :D

O, and also...
could I get a hint or a tip for another problem...

Prove or give counterexample: if U1, U2, W are subspaces of V such that V is a direct sum of U1 and W and V is a direct sum of U2 and W, then U1=U2

I haven't devoted very much time to this problem yet, I admit, but I cannot see an easy way to show it is true (my brain tells me it is...)
 
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Bruno J.

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Jun 2009
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Why not use the fact that \(\displaystyle a(bv)=(ab)v\)? If \(\displaystyle a=0\), you are done; otherwise...
 
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Oct 2008
100
9
:\
Sorry I don't really understand what you meant.

I can already show that if a=0 or v=0 then av=0, but I am having trouble showing \(\displaystyle av=0 =>\) \(\displaystyle a=0\) or \(\displaystyle v=0\)
 

Bruno J.

MHF Hall of Honor
Jun 2009
1,266
498
Canada
:\
Sorry I don't really understand what you meant.

I can already show that if a=0 or v=0 then av=0, but I am having trouble showing \(\displaystyle av=0 =>\) \(\displaystyle a=0\) or \(\displaystyle v=0\)
Suppose \(\displaystyle av=0\). If \(\displaystyle a=0\), then we are done. Otherwise, since \(\displaystyle a \in F\) and \(\displaystyle F\) is a field, we can find \(\displaystyle a^{-1}\in F\) with \(\displaystyle a^{-1}a =1\). So we have \(\displaystyle av=0 \Rightarrow a^{-1}(av)=a^{-1}(0) = 0 \Rightarrow (a^{-1}a)v=0 \Rightarrow 1v = v= 0\). So we've shown that if \(\displaystyle a \neq 0\), then \(\displaystyle v=0\); in any case, \(\displaystyle a=0\) or \(\displaystyle v=0\).
 
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Oct 2008
100
9
Nice, I get it now. Thanks!
 

HallsofIvy

MHF Helper
Apr 2005
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O, and also...
could I get a hint or a tip for another problem...

Prove or give counterexample: if U1, U2, W are subspaces of V such that V is a direct sum of U1 and W and V is a direct sum of U2 and W, then U1=U2

I haven't devoted very much time to this problem yet, I admit, but I cannot see an easy way to show it is true (my brain tells me it is...)
Let A be a basis for U1, B a basis for U2, and C a basis for W. Then both AUC and BUC are bases for W. Show that A and B span the same space.
 
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