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...)

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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