I understand what a zero vectors is in $R^n$ but I need some help visualising other zero vectors:
For example, the vector space of all functions $${ y : \mathbb R \rightarrow \mathbb R \ \ | \ y''+xy'+e^xy=0 } $$
Is the zero vector just $z(x)=0$ ? Explicit examples of less obvious vector spaces would be greatly appreciate
ted.
Another example could be the set of all functions $$y:\mathbb R\rightarrow\mathbb R \ \ | \ y''= 0$$
In this example wouldn't the zero vector be any functions $z(x)=ax+b$ but does this contradict the fact that the zero vector is unique? Or does that fact mean that the set above is NOT a vector space?
Kind Regards,
Hugh
Best Answer
Yes, the zero vector is the zero function $z(x) = 0$ (in both of your examples). In your second example, $z(x) = ax+b$ is NOT the zero element, because for a generic function $f(x)$, it is NOT true that $f(x) + ax + b = f(x)$. The "zero vector" must be an "additive identity", meaning if you add it to anything, it doesn't change.