A [[Function]] $f: X\to Y$ is [[Surjective]] if for every element $y$ in the [[Codomain]] $X$, there [[Existential Quantifier|exists]] an element $x$ in the [[Domain]] $X$ such that $f(x)=y$. For some [[Surjective]] function $f$: $\huge \forall y\in B \exists x\in A : f(x) = y $ >[!example]- >Some [[Linear Transformation]] $T: \R^n \to \R^n$ is [[Surjective|Onto]] if [[Universal Quantifier|For any]] $\vec b \in \Rn n$, there is at least one corresponding input $\vec x \in \R^n$ such that $\vec x \mapsto \vec b$. > > >Example of a not [[Surjective|Onto]] [[Matrix]]: >$A=\augmented{ccc|c}{ >1&0&-1&-1\\ >2&2&1&2\\ >0&0&0&0 >} >$ > The [[Homogeneous Equation]] $A\vec x = \vec 0$ has an [[Infinity|infinite]] number of solutions, therefore #### Composition For the [[Composition]] of two [[Function|Functions]] $f : A\to B, g :B\to C$, the composition $f\circ g$ is always [[Surjective]] if both $f$ and $g$ are [[Surjective]].