Suppose $V$ is a vector space and $x, y ∈ V$ with $x + y = x$. Is it true that $y$ must be the zero vector $0$ of $V$ ? Is there a specific example of a vector space $V$ and two such vectors $x, y ∈ V$ with $yz ≠ 0$ but $x + y = x$?
Is there any situation where $y$ is not $0$?
This was briefly mentioned in class and it was hinted that it is possible and I've thinking about it for a while but I can't come up with a specific example?
Does it count if $x = (1,1,1), y = (1,1,1)$, so $x+y = (2,2,2)$, where $(2,2,2) = (1,1,1)$ since its just a linear combination of $x$? I feel this example is weak and that $(2,2,2) = (1,1,1)$ is not an equality.
I've also been trying to think about vector spaces that are not in $R$ or some weird ones where there are only a couple elements in $V$.
Any thoughts on this? I've been stuck thinking about this for a while.
Here's a proof I did to prove $y=0$, but I don't know if it's weak. Let $x,yEV$ where $x+y=x$. So, the $i-th$ components are $x_i, y_i$. If $x_i+y_i=x_i$ then, by adding the additive inverse of $x_i$, we get $x_i+y_i+(-x_i)=x_i+(-x_i)$, so $y_i=0$. Thus, $y=0$.
Since $x+y=x$, we have $(-x)+x+y=(-x)+x=0$ so that $y=0$.