Left inverse of a linear map

A left inverse of a linear map $T:V\to W$ is a linear map $L:W\to V$ satisfying $LT=I_V$ . Therefore $T$ is injective: $Tv=0$ implies $v=LTv=0$ .

Rectangular matrices

If a real matrix $B$ has linearly independent columns, then its Gram matrix $B^TB$ is invertible and

L=(B^TB)^{-1}B^T

is a left inverse. For complex matrices, use conjugate transpose instead of transpose. The product $BL$ is the orthogonal projection onto the column space; it is not generally the identity on the larger ambient space.

Example

The inclusion $T(x,y)=(x,y,0)$ has left inverse $L(x,y,z)=(x,y)$ . This lets a vector in a two-dimensional moving plane be described by two coordinates even when the ambient space has dimension three.

References

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