Null Space

The Null Space of a linear map $T$ is denoted $\mathrm{null} \, T$ and is the set of all elements $v \in V$ that map to $0$:

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

A null space is a subspace itself:

• As $T$ is a linear map, $T(0) = 0$ and therefore $0 \in \mathrm{null} \, T$
• $T(u + v) = Tu + Tv = 0+0 = 0$, therefore $u+v \in \mathrm{null} \, T$
• $T(\lambda v) = \lambda T(v) = \lambda 0 = 0$ therefore $\lambda v \in \mathrm{null} \, T$

Injectivity

A function $T : V \to W$ is injective if $Tu = Tv$ implies that $u = v$. That is, each input maps to a unique output.

A linear map $T$ is injective if and only if $\mathrm{null} \, T = \{0\}$.

Proof:

If $T$ an injective linear map, then we must remember that $0$ always maps to $0$ (as with all linear maps) and the mapping is exclusive, so injectivity implies $\mathrm{null} \, T = \{0\}$. This is the only if condition.

For the if condition, we take $\mathrm{null} \, T = \{0\}$. We can then assume that there are two elements $u,v$ that map to the same output, that is $Tu = Tv$. We can rearrange:

$$ Tu = Tv \\ Tu - Tv = 0 \\ T(u-v)=0 $$

This implies that $u-v$ is in the null space, but the only element there is $0$, so $u-v = 0$ and $u=v$.

Range

The range of $T \in \mathcal{L}(V, W)$ is the subset of $W$ that contains all vectors mapped to by the elements of $V$.

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

The range, unsurprisingly, is also a subspace.

$T$ is called surjective if $\mathrm{range}\,T = W$, that is, under $T$ every element of $W$ is mapped to by an element of $V$.