Linear Algebra - Function (Set)

Definition

For each input element in a set <math>A</math> , a function assigns a single output element from another set <math>B</math> .

• <math>A</math> is called the domain of the function
• <math>B</math> is called the co-domain

Articles Related

Syntax

<math>f(A) = B</math>

or in Mathese:

<math>f : A \rightarrow B</math>

Example

• The function that doubles its input:

f({1, 2, 3,...}) = {(1,2),(2,4),(3,6),(4,8),....}

• The function that multiplies the numbers forming its input:

f({1, 2, 3,...}x{1, 2, 3,...}) = {((1,1),2),((1,2),4),...,((2,2),4),(2,3),6), ....}

• Caesar’s Cryptosystem

Each letter is mapped to one three places ahead, wrapping around, so MATRIX would map to PDWULA. The function mapping letter to letter can be written as:

{('A','D'),('B','E'),('C','F'),('D','G'),...,('W','Z'),('X','A'),('Y','B'),('Z','C')} 

Both the domain and the co-domain are {A,B, …,Z}

Image

for a function <math>f : V \mapsto W</math> , the image of f is the set of all images of elements of the domain: <MATH> {f (v) : v \in V} </MATH>

The image of a function is the set of all images of inputs. Mathese:

lm f

The output of a given input is called the image of that input. The image of A under the function f is denoted f(A).

There might be elements of the co-domain that are not images of any elements of the domain.

Set

Set of functions: For sets F and D, <math>F^D</math> denotes all functions from D to F.

For finite sets, the cardinality <math>|F^D| = |F|^{|D|}</math>

Function

Identity

Function - Identity

Composition

Linear Algebra - Function composition

Inverse

Functions f and g are functional inverses if

f circ g

and

g circ f

are defined and are identity functions.

Invertible

A function that has an inverse is invertible.

Invertible functions are:

• one-to-one
• onto

Function Invertibility Theorem: A function f is invertible if and only if it is one-to-one and onto.

Linear-Function Invertibility Theorem: A function <math>f: V \mapsto W</math> is invertible iff

• <math>\text{dim Ker f = 0 and dim lm f = dim W }</math>
• <math>\text{dim Ker f = 0 and dim V = dim W }</math> (For f to be invertible, need dim V = dim W)

Extracting an invertible function:

Let <math>A = \begin{bmatrix} \begin{array}{r|r|r} 1 & 2 & 1 \\ 2 & 1 & 1 \\ 1 & 2 & 1 \\ \end{array} \end{bmatrix} </math> and define <math>f : \mathbb{R}^3 \mapsto \mathbb{R}^3</math> by <math>f (x) = Ax</math>

Define <math>W^* = lm f = Col A = Span \{[1, 2, 1], [2, 1, 2], [1, 1, 1]\}</math>

One basis for <math>W^*</math> is:

• <math>w_1 = [0, 1, 0]</math> ,
• <math>w_2 = [1, 0, 1]</math>

Pre-images for <math>w_1</math> and <math>w_2</math> :

• <math>v_1 = [\frac{1}{2},−\frac{1}{2},\frac{1}{2}]</math>
• and <math>v_2 = [−\frac{1}{2},\frac{1}{2},\frac{1}{2}]</math>

for then <math>A.v_1 = w_1</math> and <math>A.v_2 = w_2</math>

Let <math>V^* = Span \{v_1, v_2\}</math> Then <math>f : V^* \mapsto lm f </math> is onto and one-to-one and therefore invertible.

Linear

Linear Algebra - Linear Function (Weighted sum)

Relationship

One-to-one

<math>f : V \rightarrow W</math> is one-to-one if f (x) = f (y) implies x = y

One-to-One Lemma: A linear function is one-to-one if and only if its kernel is a trivial vector space. Equivalent: if its kernel has dimension zero.

f is one-to-one iff <math>\href{dimension}{dim} \text{ } \href{#kernel}{Ker} \text{ } f = 0</math>

Onto

<math>f : V \rightarrow W</math> is onto if for every <math>z \in F</math> there exists an a such that <math>f(a) = z</math>

f is onto if its image equals its co-domain

For any linear function <math>f : V \mapsto W</math> , f is onto if <math>\href{dimension}{dim} \text{ } \href{#image}{lm} f = \href{dimension}{dim} \text{ } W</math>