# Simple Vector Space Proof

#### billa

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 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.

MHF Hall of Honor
Why not use the fact that $$\displaystyle a(bv)=(ab)v$$? If $$\displaystyle a=0$$, you are done; otherwise...

• billa

#### billa

:\
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
:\
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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• billa

#### billa

Nice, I get it now. Thanks!

#### HallsofIvy

MHF Helper
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.

• billa and Bruno J.