$\forall$ = for all

$\in$ = belong to .

$\exists$= there exists.

$\Longrightarrow$= implies

$\Longleftrightarrow$ = it is equivalent to

$\mathbb Z$= set of integers.

$\mathbb R$= set of real numbers.

$\mathbb Q$= set of rational numbers

$\mathbb C$= set of complex numbers

$\phi$= empty set

$M_{n \times n}(\mathbb R)$= set of $n \times n$-real matrices

$\text{GL}_n (\mathbb R)$= set of $n \times n$-real $invertible$ matrices

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Some facts on functions:

A function (map) $f: A\to B$ is a rule which assigns each $x \in A$ to a unique $f(x) \in B$. $A$ is called $domain$ of $f$. $f(x)$ is called the $image$ of $x$, while $x$ is called a $pre-image$ of $f(x)$. The $range$ of $f$ is the set of all image

\[ Range(f): = \{f(x)| \forall x \in A \} \subset B. \]

$f$ is called $injective$ if $f(x)= f(y)$ then $x= y$ for all such $x, y \in A$.

$f$ is called $surjective$ if $Range(f) = B$, equivalently, $\forall y \in B$, $\exists x \in A$ so that $f(x)= y$.

$f$ is called $bijiective$ if it is injetive and surjective.

Let $f ; A \to B$ and $g : B \to C$ be two functions. the $composite$ of $g$ and $f$ is a function giiven by \[ g \circ f : A \to C, \ \ (g\circ f)(x)= g(f(x)). \] Let $f , g : A \to B$ be functions. $f = g$ means $f(x)= g(x), \forall x \in A$

Let $f: A\to B$ be a function. The function $g: B\to A$ is called an $inverse$ of $f$ if $ g\circ f = I_A$ and $f\circ g= I_B$. Here $I_A: x \to x$ is the $identity$ function. We write $g = f^{-1}$

$f$ admits an inverse if and only if $f$ is bijecive.