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1.11 Linear operators

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Definition 1

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

  1. L ( α x ) = α L ( x ) ∀ x ∈ X , α ∈ K
  2. L ( x 1 + x 2 ) = L ( x 1 ) + L ( x 2 ) ∀ x 1 , x 2 ∈ X .

We call such transformations linear operators .

  • X = R N , Y = R M L : R N → R M is an M × N matrix
  • Fourier transform: F ( x ( t ) ) = ∫ - ∞ ∞ x ( t ) e - j w t d t F : L 2 ( R ) → L 2 ( R )

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

R ( L ) = { L ( x ) ∈ Y : x ∈ X } .

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

N ( L ) = { x ∈ X : L ( x ) = 0 } .

If L is linear, then both R ( L ) and N ( L ) are subspaces.
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Source:  OpenStax, Digital signal processing. OpenStax CNX. Dec 16, 2011 Download for free at http://cnx.org/content/col11172/1.4
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