= Solution
For a <linear map> $T:V\to W$, its <image of a linear map> and <kernel of a linear map> are
$$
\boxed{\operatorname{im}T=\{T(v):v\in V\}\subseteq W,\qquad \ker T=\{v\in V:T(v)=0_W\}\subseteq V.}
$$
Their <dimensions> define the <rank of a linear map> and <nullity>:
$$
\boxed{\operatorname{rank}T=\dim(\operatorname{im}T),\qquad\operatorname{nullity}T=\dim(\ker T).}
$$
The <image of a linear map> measures which output <vectors> can occur, while the <kernel of a linear map> measures which input <vectors> are sent to zero.
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