1.11 Linear operators

Digital signal processing Page 1 / 1

Definition 1

A transformation (mapping)

L : X \to Y

from a vector space

X

to a vector space

Y

(with the same scalar field

K

) is a linear transformation if:

We call such transformations linear operators .

$X = R^{N}$ , $Y = R^{M}$ $L : R^{N} \to R^{M}$ is an $M \times N$ matrix
Fourier transform: $F (x (t)) = \int_{- \infty}^{\infty} x (t) e^{- j w t} d t$ $F : L_{2} (R) \to L_{2} (R)$

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

R (L) = {L (x) \in Y : x \in X} .

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

N (L) = {x \in 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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