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NF02 committed Dec 5, 2024
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1 change: 1 addition & 0 deletions Primo anno/Calcolatori elettronici/book.tex
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Expand Up @@ -289,4 +289,5 @@ \subsection{Dischi meccanici}
comunque essendo un apparecchio meccanico quindi più ci si avvicina al centro
del disco più i settori sono meno distanti tra loro.


\end{document}
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1,111 changes: 0 additions & 1,111 deletions Primo anno/algebra e geometria/#algebra-e-geometria.tex#
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1 change: 0 additions & 1 deletion Primo anno/algebra e geometria/.#algebra-e-geometria.tex
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9 changes: 6 additions & 3 deletions Primo anno/algebra e geometria/algebra-e-geometria.aux
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28 changes: 16 additions & 12 deletions Primo anno/algebra e geometria/algebra-e-geometria.log
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1 change: 1 addition & 0 deletions Primo anno/algebra e geometria/algebra-e-geometria.out
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Expand Up @@ -26,3 +26,4 @@
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35 changes: 35 additions & 0 deletions Primo anno/algebra e geometria/algebra-e-geometria.tex
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Expand Up @@ -1106,4 +1106,39 @@ \subsection{Iniettività di applicazioni lineari}
proprietà e automaticamente varrà anche l'altra.\\
Ora, supponendo di aver determinato grazie ai risultati precedenti che un'applicazione lineare
$f$ data è biiettiva.

\section{Composizione di applicazioni, inversa e prodotto di matrici}
\label{sec:Compinveeproddimatrici}

Ricordiamo che, data una funzione $f:X\to Y$ tra due insiemi, questa si
dice \textit{invertibile} se esiste una funzione $g:Y\to X$ (detta
appunto l'inversa di $f$) tale che
\begin{equation}
\label{eq:Compinveeproddimatrici1}
f \circ g=id_{y}, \text{ } g\circ f=id_{x}
\end{equation}
e si denota che con \textit{id} la funzione che manda ogni elemento in se
stesso\footnote{Più precisamente, $id_{x}$ è la funzione $X\to X$ che
manda ogni elemento di $x$ in se stesso e $id_{y}$ denota la funzione
$Y\to Y$ che manda ogni elemento di $Y$ in se stesso.} e con il simbolo
$\circ$ la composizione di funzioni, ovvero l'operazione che consiste
nell'applicare prima una fuzione e poi l'altra: più precisamente,
ricordiando che ogni volta che si hanno due funzioni $f:X\to Y$ e
$g:Y\to Z$, tali che \textit{il codominio della prima coincida con il
della seconda,} allora, per ogni elemento che $Y$ è anche, per ogni
elemento $x\in X$, si può applicare prima $f$ ottenendo $f(x)\in Y$,
e poi dal momento che $Y$ è anche il dominio della $g$ si può applicare
la $g$ a $f(x)$. In questo modo si ottiene una nuova funzione che
associa a ogni elemento di $X$ un elemento di $Z$:
\begin{equation*}
\underset{x\to g(f(x))}{f:X\to Z}
\end{equation*}
Quindi, le (\ref{eq:Compinveeproddimatrici1}) significano che una
funzione $f:X\to Y$ è invertibile se esiste una funzione $g:Y\to X$
tale $g(f(x))=x$ $x\in X$ e $f(g(y))=y$ per ogni $y\in Y$.\\
Ora, si può vedere che le uniche funzioni di $f$ invertibili sono
quelle biiettive. Ad esempio, considerando $X=\{1,2,3\}$, $Y=\{a,b,c\}$
e la funzione $f:X\to Y$ biiettiva, che ha come inversa la funzione
$g:Y\in X$ rappresentata nella figura:

\end{document}
1 change: 1 addition & 0 deletions Primo anno/algebra e geometria/algebra-e-geometria.toc
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