First isomorphism theorem for linear maps

Statement

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a linear map. Then the map $$\bar T \colon V / \ker T \to \im T, \qquad v + \ker T \mapsto T(v),$$ is well defined, and it is a linear isomorphism. Here $V / \ker T$ is the quotient space by the kernel, and $\im T$ is the image of $T$.

Remarks

In finite dimensions, it gives the rank–nullity theorem: $\dim V - \dim \ker T = \dim \im T$. The same theorem holds for groups, rings and modules. For a surjective map $T$, it says $V / \ker T \cong W$.

Explanations

L1 human

A linear map identifies vectors that differ by an element of the kernel. After we glue these vectors together, the map becomes one to one. So the quotient by the kernel is a copy of the image.