Kernel and Image of a Linear Map

In the setting of Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, let $K$, with $+$, $\cdot$, $0$ and $1$, be a field, let $V$ and $W$, each with its addition and scalar multiplication, be vector spaces over $K$, let $0_{W}$ be the zero vector of $W$, and let $T$ be a linear map from $V$ to $W$.

(i) Kernel. The kernel of $T$ is the subset

$$\ker T=\{v\in V:T(v)=0_{W}\}$$

of $V$ (a set by Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper (i)).

(ii) Image. The image of $T$ is the image

$$\operatorname{im}T=T[V]$$

of $V$ under $T$ (a subset of $W$: it is the range of $T$ by Basic Properties of Functions: Equality, Composition, Identity, Inverse and Restriction (v), and $\operatorname{ran}T\subseteq W$ by Functions, Values of a Function, and Functions from One Class to Another (iii)).