Right inverse of a linear map

A right inverse of a linear map $T:V\to W$ is a linear map $R:W\to V$ such that

TR=I_W.

It supplies a solution $v=Rw$ to $Tv=w$ for every $w\in W$ , so $T$ is surjective. A right inverse need not be unique: adding a linear map $K:W\to V$ satisfying $TK=0$ preserves the identity.

Compatibility and regularity

If a differential operator is inverted only on a subspace of compatible sources, that subspace is the target $W$ in the right-inverse statement. Existence of an algebraic right inverse does not assert boundedness, smooth parameter dependence or support preservation; each is an additional property of the chosen inverse.

Example

For $T(x,y)=x$ , the map $R(t)=(t,0)$ is a right inverse. Here $RT$ is a projection, not the identity on $\mathbb R^2$ .

References

Sheldon Axler, Linear Algebra Done Right, 4th ed., Chapters 2–3, 6 and 9.