Surjectivity

Surjective

A function $T:V\to W$ is called surjective if its range equals $W$.

Also called onto.

To illustrate the definition above, note that of the range examples we saw, only the differentiation map is surjective. The zero map is surjective in the special case $W=\{0\}$.

Whether a linear map is surjective depends on what we are thinking of as the vector space into which it maps.

For example, the differentiation map $D\in \mathcal{L}({\mathcal{P}}_5(\mathbb{R}))$ defined by $Dp=p^{\prime }$ is not surjective, because the polynomial $x^5$ is not in the range of $D$; there are no fifth-degree polynomials whose derivative are also fifth-degree polynomials, so ${\mathcal{P}}_5(\mathbb{R})$ is too big of an output space to be surjective.

However, the differentiation map $S\in \mathcal{L}({\mathcal{P}}_5(\mathbb{R}),{\mathcal{P}}_4(\mathbb{R}))$ defined by $Sp=p^{\prime }$ is surjective, because its range equals ${\mathcal{P}}_4(\mathbb{R})$, which is the vector space into which $S$ maps.