Range

Range

For $T\in \mathcal{L}(V,W)$, the range of $T$ is the subset of $W$ consisting of those vectors that are equal to $Tv$ for some $v\in V$:

$$\text{range }T=\{Tv:v\in V\}$$

The range is essentially the set of outputs of a linear map.

Examples

• If $T$ is the zero map from $V$ to $W$, meaning that $Tv=0$ for every $v\in V$, then $\text{range }T=\{0\}$.

• Suppose $T\in \mathcal{L}({\mathbb{R}}^2,{\mathbb{R}}^3)$ is defined by $T(x,y)=(2x,5y,x+y)$. Then

$$\text{range }T=\{(2x,5y,x+y):x,y\in \mathbb{R}\}$$

Note that $\text{range }T$ is a subspace of ${\mathbb{R}}^3$. We will soon see that the range of each element of $\mathcal{L}(V,W)$ is a subspace of $W$.

• Suppose $D\in \mathcal{L}(\mathcal{P}(\mathbb{R}))$ is the differentiation map defined by $Dp=p^{\prime }$. Because for every polynomial $q\in \mathcal{P}(\mathbb{R})$ there exists a polynomial $p\in \mathcal{P}(\mathbb{R})$ such that $p^{\prime }=q$, the range of $D$ is $\mathcal{P}(\mathbb{R})$.