# Connexions

You are here: Home » Content » Digital Signal Processing » Linear Operators

### Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection (Course):

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

# Linear Operators

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

## Definition 1

A transformation (mapping) L:XYL:XY 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)xXxX, αKαK
2. L(x1+x2)=L(x1)+L(x2)L(x1+x2)=L(x1)+L(x2)x1,x2Xx1,x2X.

We call such transformations linear operators.

## Example 1

• X=RNX=RN, Y=RMY=RML:RNRML:RNRM 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:XYL:XY 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.

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks