Basic Definitions of Mappings

Mapping and Domain

$U$ be a set such that $U\subseteq E$ The relation $f$ is said mapping when $\forall u\in U : \exists!w\in F \ni`f(u)=w$ Said domain and is indicated with $dom(f)$

A vector $w$ such that $w=f(u)$ is said to be the mapped (or transformed) of $u$ through $f$

Image

Image ($Im$) is a set of all all vectors $w$ that exist such that $f(u)=w$ $Im(f)=\{w\in F | \exist u\in E\ni`f(u)=w\}$

dom/domain = Input Image = output $\to$ also means a subset

Surjective, injective, bijective

Mapping $f$ is surjective if the image of $f$ coincides with $F:Im(f)=F$

Mapping $f$ is injective if $\forall u,v \in E$ with $f(u)=f(v)\to u=v$ (if $u$ is different from $v$, then so will $f(u)/f(v)$)

Mapping $f$ is bijective if $f$ is injective and surjective

Linear Mappings

Linear Mapping

Linear mapping if the following properties are valid:

• Additivity: $\forall u,v \exists E:f(u+v)=f(u)+f(v)$
• Homogeneity: $\forall\lambda \in K$ and $\forall v\in E :f(\lambda v)= \lambda f(v)$

(This happens very very rarely)

Affine Mapping

Said affine if $ g(v) = f(v) - f(u)$ is linear

Linear Mappings

Image of $U$ through $f$, $f(U)$ is the set $f(U)=\{w\in F | \exist u \in U \ni`f(u)=w\}$ Follows that the triple $(f(u),+,.)$ is a vectors subspace of $(F,+,.)$

Inverse Image

$E\to W\subset F$ The inverse image of $W$ through $f$, indicated with $f^{-1}(w)$ is a set defined as: $f^{-1}w=\{u\in E|f(u)\in W\}$

Matrix Representation

Linear mapping can be expressed as the product of a matrix by a vector $y=f(x)=Ax$

Image from a matrix

The mapping $y=f(x)$ is expressed as a matrix equation $y=Ax$. Follows that the image of the mapping $Im(f)$ is spanned by the column vectors of the matrix A: $Im(f)=L(a^1,a^2,...a^n)$ where $A=(a^1,a^2,...a^n)$

(Just convert matrix into (A). Columns into own (C))

Endomorphisms and Kernel

Endomorphism

$E\to F$. If $E=F$, $f:E\to E$ then linear mapping is endomorphism

Null Mapping

$O:E \to F$ is a mapping defined as $\forall v \in E: O(v) = O_F$

Identity Mapping

(Should be same dimension) $I:E\to F$ is a mapping defined as $\forall v\in E: I(v)=v$

Matrix Representation

$E\to E$ be endomorphism. Mapping of multiplication of a square matrix by a vector: $f(x):Ax$

$y=f(X)=Ax$. The inverse function $x=f^{-1}(y)=A^{-1}y$

Linear dependence

Needs to be endomorphism $v_1...$ likely dependent then so are $f(v)...$

Kernal

$E\to F$ linear mapping $ker(f)=\{v\in E| f(v) = o_F$

Kernal as Vector Space

The triple $(Ker(f),+,.)$ is a vector subspace of $(E,+,.)$

Kernal and Injection

$f:E\to F$ be a linear mapping $u,v\in E$. Follows that $f(u)=f(v)$ if and only if $u-v\in Ker(f)$

Theorem

Mapping is injective if and only if: $Ker(f)=\{o_e\}$

Linear Independence and injection

$E\to F$ $V_1.....$ be $n$ linearly independent vectors $\in E$. If $f$ is injective then $f(v_1)...$ are also linearly independent vectors of $f$ Endomorphism $f$ is invertible if and only if it is injective