MAT3040 Advanced Linear Algebra ·

Statement

Let T:V→VT:V\to V be a linear transformation with dim⁡V<∞\dim V<\infty. Then V=ker⁡T⊕im⁡TV=\ker T\oplus\operatorname{im}T.

Proposed proof

Since both the domain and codomain are VV, we have ker⁡(T)≤Vandim⁡(T)≤V. \ker(T)\leq V \qquad\text{and}\qquad \operatorname{im}(T)\leq V.

In the proof of the Rank--Nullity Theorem, we have ker⁡(T)⊕U=V, \ker(T)\oplus U=V, where U≅T(U)=im⁡(T). U\cong T(U)=\operatorname{im}(T). Therefore one has ker⁡(T)⊕im⁡(T)=V. \ker(T)\oplus\operatorname{im}(T)=V.