Null Space

Null space

For $T\in \mathcal{L}(V,W)$, the null space of $T$, denoted by null $T$, is the subset of $V$ consisting of those vectors that $T$ maps to $0$:

$$\text{null }T=\{v\in V:Tv=0\}$$

This means that null spaces are specific to a linear map acting on a subspace, not on the subspace itself!

Examples

• If $T$ is the zero map from $V$ to $W$, meaning that $Tv=0$ for every $v\in V$, then $\text{null }T=V$.
• Suppose $\varphi \in \mathcal{L}({\mathbb{C}}^3,\mathbb{C})$ is defined by $\varphi (z_1,z_2,z_3)=z_1+2z_2+3z_3$. Then null $\varphi$ equals $\{(z_1,z_2,z_3)\in {\mathbb{C}}^3:z_1+2z_2+3z_3=0\}$, which is a subspace of the domain of $\varphi$ (see Null Space is a Subspace)
• Suppose $D\in \mathcal{L}(\mathcal{P}(\mathbb{R}))$ is the differentiation map defined by $Dp=p^{\prime }$. The only functions whose derivative equals the zero functions are the constant functions. Thus, the null space of $D$ equals the set of constant functions.
• Suppose that $T\in \mathcal{L}(\mathcal{P})(\mathbb{R})$ is the multiplication by $x^2$ map defined by $(Tp)(x)=x^{2}p(x)$. The only polynomial $p$ such that $x^{2}p(x)=0$ for all $x\in \mathbb{R}$ is the $0$ polynomial. Thus, $\text{null }T=\{0\}$.
• Suppose $T\in \mathcal{L}({\mathbb{F}}^{\infty })$ is the backward shift defined by $T(x_1,x_2,x_3,\dots )=(x_2,x_3,\dots )$. Then, $T(x_1,x_2,x_3,\dots )$ equal $0$ if and only if the numbers $x_2,x_3,\dots$ are all $0$. Thus, $\text{null }T=\{(a,0,0,\dots ):a\in \mathbb{F}\}$.