Linear Algebra - Function (Set)

1 - Definition

For each input element in a set

$A$

, a function assigns a single output element from another set

$B$

.

• $A$

is called the domain of the function

• $B$

is called the co-domain

3 - Syntax

$f(A) = B$

or in Mathese:

$f : A \rightarrow B$

4 - 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}

5 - Image

for a function

$f : V \mapsto W$

, the image of f is the set of all images of elements of the domain:

$${f (v) : v \in V}$$

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.

Example
A function f : {1, 2, 3, 4} −> {'A','B','C','D','E'}
The image of f is lm f = {'A','B','C','E'}
'D' is in the co-domain but not in the image

6 - Set

Set of functions: For sets F and D,

$F^D$

denotes all functions from D to F.

For finite sets, the cardinality

$|F^D| = |F|^{|D|}$

7 - Function

7.3 - Inverse

Functions f and g are functional inverses if and are defined and are identity functions.

7.4 - Invertible

A function that has an inverse is invertible.

Invertible functions are:

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

$f: V \mapsto W$

is invertible iff

• $\text{dim Ker f = 0 and dim lm f = dim W }$
• $\text{dim Ker f = 0 and dim V = dim W }$

(For f to be invertible, need dim V = dim W)

Extracting an invertible function:

Let

$A = \begin{bmatrix} \begin{array}{r|r|r} 1 & 2 & 1 \\ 2 & 1 & 1 \\ 1 & 2 & 1 \\ \end{array} \end{bmatrix}$

and define

$f : \mathbb{R}^3 \mapsto \mathbb{R}^3$

by

$f (x) = Ax$

Define

$W^* = lm f = Col A = Span \{[1, 2, 1], [2, 1, 2], [1, 1, 1]\}$

One basis for

$W^*$

is:

• $w_1 = [0, 1, 0]$

,

• $w_2 = [1, 0, 1]$

Pre-images for

$w_1$

and

$w_2$

:

• $v_1 = [\frac{1}{2},−\frac{1}{2},\frac{1}{2}]$
• and $v_2 = [−\frac{1}{2},\frac{1}{2},\frac{1}{2}]$

for then

$A.v_1 = w_1$

and

$A.v_2 = w_2$

Let

$V^* = Span \{v_1, v_2\}$

Then

$f : V^* \mapsto lm f$

is onto and one-to-one and therefore invertible.

7.6 - Relationship

7.6.1 - One-to-one

$f : V \rightarrow W$

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

$\href{dimension}{dim} \text{ } \href{#kernel}{Ker} \text{ } f = 0$

7.6.2 - Onto

$f : V \rightarrow W$

is onto if for every

$z \in F$

there exists an a such that

$f(a) = z$

f is onto if its image equals its co-domain

For any linear function

$f : V \mapsto W$

, f is onto if

$\href{dimension}{dim} \text{ } \href{#image}{lm} f = \href{dimension}{dim} \text{ } W$