Onto

Let $f$ be a Function defined on a Set $S$ and taking values in a set $T$. Then $f$ is said to be onto (a.k.a. a Surjection) if, for any $t\in T$, there exists an $s\in S$ for which $t=f(s)$.

Let the function be an Operator which Maps points in the Domain to every point in the Range and let $\Bbb{V}$ be a Vector Space with ${\bf X}, {\bf Y}\in \Bbb{V}$. Then a Transformation $T$ defined on $\Bbb{V}$ is onto if there is an ${\bf X} \in \Bbb{V}$ such that $T({\bf X}) = {\bf Y}$ for all ${\bf Y}$.

See also Bijection, One-to-One