Worksheet 17

1. Given a linear transformation $T : V \to W$, the null space of $T$ is \(\mathcal Z(T) := \lbrace v \in V \mid T(v) = 0 \rbrace\) Show that $T$ is injective if and only $\mathcal Z(T) = 0$.

2. Given a linear transformation $T: V \to W$, the range of $T$ is \(\mathcal R(T) := \lbrace w \in W \mid w = T(v) \text{ for some } v \in V \rbrace\) Show that $T$ is surjective if and only if $\mathcal R(T) = W$.

3. Show that a linear transformation $T: V \to W$ is a bijection if and only $\mathcal Z(T) = 0$ and $\mathcal R(T) = W$.