Definition:Index of Linear Transformation

Definition

Let $U, V$ be vector spaces over a field $K$.

Let $T: U \to V$ be a linear transformation of finite index.

The index of $T$ is defined as:

$\map {\mathrm{ind} } T := \map \dim {\map \ker T} - \map {\mathrm {codim}} {\Img T}$

where:

$\map \dim {\map \ker T}$ denotes the dimension of the kernel

$\map {\mathrm {codim}} {\Img T}$ denotes the codimension of image in $V$

Also see

• Linear Transformation has Finite Index iff Pseudoinverse exists
• Definition:Pseudoinverse of Linear Transformation

Sources

• 2002: Peter D. Lax: Functional Analysis: Chapter $27$: Index Theory