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Linear Operators

Module by: Mark A. Davenport. E-mail the author

Definition 1

A transformation (mapping) L:X→YL:X→Y from a vector space XX to a vector space YY (with the same scalar field KK) is a linear transformation if:

  1. L(αx)=αL(x)L(αx)=αL(x)∀x∈X∀x∈X, α∈Kα∈K
  2. L(x1+x2)=L(x1)+L(x2)L(x1+x2)=L(x1)+L(x2)∀x1,x2∈X∀x1,x2∈X.

We call such transformations linear operators.

Example 1

  • X=RNX=RN, Y=RMY=RML:RN→RML:RN→RM is an M×NM×N matrix
  • Fourier transform: F(x(t))=∫-∞∞x(t)e-jwtdtF(x(t))=∫-∞∞x(t)e-jwtdtF:L2(R)→L2(R)F:L2(R)→L2(R)

Let L:X→YL:X→Y be an operator (linear or otherwise). The range spaceR(L)R(L) is

R ( L ) = { L ( x ) ∈ Y : x ∈ X } . R ( L ) = { L ( x ) ∈ Y : x ∈ X } .
(1)

The null spaceN(L)N(L), also known as “kernel”, is

N ( L ) = { x ∈ X : L ( x ) = 0 } . N ( L ) = { x ∈ X : L ( x ) = 0 } .
(2)

If LL is linear, then both R(L)R(L) and N(L)N(L) are subspaces.

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