Knowlpedia

Isomorphism theorem for linear operators

The image of a linear map is isomorphic to the quotient by its kernel

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

im⁡T≅X/ker⁡T.\operatorname{im}T \cong X/\ker T.

In particular, if dimensions are finite,

dim⁡(im⁡T)=codim⁡(ker⁡T)=dim⁡(X/ker⁡T).\dim(\operatorname{im}T)=\operatorname{codim}(\ker T)=\dim(X/\ker T).
Remarks

Context. This theorem explains that a linear map is completely determined (up to isomorphism) by its . The quotient here is the formed by collapsing ker⁡T\ker T to 00.

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