Kernel and Image of a Linear Map Are Subspaces

In the setting of Linear Algebra: Standing Notation, let $K$ be a field, let $V$ and $W$ be vector spaces over $K$, and let $T\in\mathcal{L}(V,W)$.

(i) Linear combinations. $T0=0$ and $T(-v)=-Tv$ for every $v\in V$; and for $n\in\mathbb{N}_{0}$, $\lambda_{1},\dots,\lambda_{n}\in K$ and $v_{1},\dots,v_{n}\in V$,

$$T\Bigl(\sum_{i=1}^{n}\lambda_{i}v_{i}\Bigr)=\sum_{i=1}^{n}\lambda_{i}\,Tv_{i}.$$

(ii) Subspaces. $\ker T$ is a linear subspace of $V$, and $\operatorname{im}T$ is a linear subspace of $W$.

(iii) Injectivity. $T$ is injective if and only if $\ker T=\{0\}$.

(iv) Surjectivity. $T$ is surjective if and only if $\operatorname{im}T=W$.