Isomorphism theorem for linear operators

Theorem (First isomorphism theorem; dimension formula). Let $T:X\to Y$ be a linear operator. Then

In particular, if dimensions are finite,

Context. This theorem explains that a linear map is completely determined (up to isomorphism) by its kernel and image . The quotient here is the quotient vector space formed by collapsing $\ker T$ to $0$.

Proof sketch. Define $\widetilde{T}:X/\ker T\to \operatorname{im}T$ by $\widetilde{T}(x+\ker T)=T(x)$. This is well-defined because $x-x'\in\ker T$ implies $T(x)=T(x')$. It is linear, surjective by definition of $\operatorname{im}T$, and injective because $\widetilde{T}(x+\ker T)=0$ forces $x\in\ker T$.