diff --git a/html/algoritmi.html b/html/algoritmi.html
index 1c8b7bb..56d0c81 100644
--- a/html/algoritmi.html
+++ b/html/algoritmi.html
@@ -285,9 +285,9 @@
r_n \mid d
-- \left \{ \begin{array}{l} r_{n + 1} = 0 \\
-0 \equiv r_{n + 1} \equiv r_{n - 1} \ (\bmod \ r_n) \iff \exists q_n \in
-\mathbb{Z} \mid r_{n- 1} = r_nq_n \end{array} \iff r_n \mid r_{n -
+
- \left \{ \begin{array}{l} r_{n + 1} = 0
+\\ 0 \equiv r_{n + 1} \equiv r_{n - 1} \ (\bmod \ r_n) \iff \exists q_n
+\in \mathbb{Z} \mid r_{n- 1} = r_nq_n \end{array} \iff r_n \mid r_{n -
1}\right.
- r_n \equiv r_{n - 2} \ (\bmod r_{n - 1})
\iff \exists q_{n - 1} \in \mathbb{Z} \mid r_n =r_{n - 1} q_{n - 1} +
@@ -364,11 +364,11 @@
- \left \{ \begin{array}{l} (p-1) \mid
\lambda(n) \\ (q - 1) \mid \lambda(n) \end{array} \right. \implies
-\exists i, j \in \mathbb{Z} \mid \lambda(n) = (p-1) \cdot i = (q - 1)
+\exists i, j \in \mathbb{Z} \mid \lambda(n) = (p-1) \cdot i = (q - 1)
\cdot j
- \textrm{MCD}(m, n) = 1 \implies p \nmid m
\land q \nmid m \implies \left \{ \begin{array}{l}p \nmid m \implies m^p
-\equiv m \iff m^{p-1} \equiv 1 \implies m^{\lambda(n)} \equiv
+\equiv m \iff m^{p-1} \equiv 1 \implies m^{\lambda(n)} \equiv
m^{(p-1)\cdot i} \equiv 1 \ (\bmod \ p) \\ q \nmid m \implies m^q \equiv
m \iff m^{q-1} \equiv 1 \implies m^{\lambda(n)} \equiv m^{(q-1)\cdot j}
\equiv 1 \ (\bmod \ q) \end{array} \right. \iff m^{\lambda(n)} \equiv 1
@@ -407,7 +407,7 @@
- Alg
- \forall i \in [0, n] \quad p_i(x) :=
-\displaystyle \prod_{\begin{subarray}{c}0 \le j \le n \\ i \neq\ j
+\displaystyle \prod_{\begin{subarray}{c}0 \le j \le n \\ i \neq\ j
\end{subarray}}{\dfrac{x - b_j}{b_i - b_j}}
- p(x) := c_0p_0(x) + \ldots + c_n
p_n(x)
diff --git a/html/coefficienti-binomiali.html b/html/coefficienti-binomiali.html
index 2eef064..1a0d525 100644
--- a/html/coefficienti-binomiali.html
+++ b/html/coefficienti-binomiali.html
@@ -360,15 +360,12 @@
class="math inline">\left[a_{1}\right]^{p}=\left[a_{1}\right]^{p}
per dimostrazione precedente
- n>1
-\implies\left(\left[a_{1}\right]+\ldots+\left[a_{n}\right]+\left[a_{n+1}\right]\right)^{p}=
-\left[a_{1}\right]^{p}+\ldots+\left[a_{n}\right]^{p}+\left[a_{n+1}\right]^{p}
+\implies\left(\left[a_{1}\right]+\ldots+\left[a_{n}\right]+\left[a_{n+1}\right]\right)^{p}= \left[a_{1}\right]^{p}+\ldots+\left[a_{n}\right]^{p}+\left[a_{n+1}\right]^{p}
- per ipotesi induttiva, \left[a_{1}\right]^{p}+\ldots+\left[a_{n}\right]^{p}+\left[a_{n+1}\right]^{p}=
-\left(\left[a_{1}\right]+\ldots+\left[a_{n}\right]\right)^{p}+\left[a_{n+1}\right]^{p}
+class="math inline">\left[a_{1}\right]^{p}+\ldots+\left[a_{n}\right]^{p}+\left[a_{n+1}\right]^{p}= \left(\left[a_{1}\right]+\ldots+\left[a_{n}\right]\right)^{p}+\left[a_{n+1}\right]^{p}
- allora, ancora per ipotesi induttiva \left(\left[a_{1}\right]+\ldots+\left[a_{n}\right]\right)^{p}+\left[a_{n+1}\right]^{p}=
-\left(\left[a_{1}\right]+\ldots+\left[a_{n+1}\right]\right)^{p}
+class="math inline">\left(\left[a_{1}\right]+\ldots+\left[a_{n}\right]\right)^{p}+\left[a_{n+1}\right]^{p}= \left(\left[a_{1}\right]+\ldots+\left[a_{n+1}\right]\right)^{p}
diff --git a/html/determinante.html b/html/determinante.html
index 4dc83ab..290c531 100644
--- a/html/determinante.html
+++ b/html/determinante.html
@@ -207,7 +207,7 @@
\rightarrow W:(v_1, \ldots, v_n) \rightarrow w
f è detta
multilineare \iff \forall i
-\in [1, n], v_1 , \ldots, v_n \in V_1 \times \ldots \times V_n, v_i,
+\in [1, n], v_1 , \ldots, v_n \in V_1 \times \ldots \times V_n, v_i,
v_i' \in V_i, \lambda, \mu \in \mathbb{K} \quad f(v_1, \ldots,
\lambda v_i+\mu v_i', \ldots, v_n) = \lambda f(v_1, \ldots, v_i,
\ldots, v_n) + \mu f(v_1, \ldots, v_i', \ldots, v_n)
@@ -316,16 +316,16 @@
class="math inline">\det(A_1, \ldots, A_i + A_j, \ldots, A_j + A_i,
\ldots, A_n) = 0
allora, per multilinearità di \det
-si ha che \det si ha che 0 =\det(A_1, \ldots, A_i + A_j, \ldots, A_j + A_i,
-\ldots, A_n) = \det(A_1, \ldots, A_i, \ldots, A_j + A_i, \ldots, A_n) +
-\det(A_1, \ldots, A_j, \ldots, A_j + A_i, \ldots, A_n) =\det(A_1,
-\ldots, A_i, \ldots, A_j, \ldots, A_n)+\det(A_1, \ldots, A_i, \ldots,
-A_i, \ldots, A_n) + \det(A_1, \ldots, A_j, \ldots,A_j, \ldots, A_n) +
-\det(A_1, \ldots, A_j, \ldots, A_i, \ldots, A_n) = \det(A_1, \ldots,
-A_i, \ldots, A_j, \ldots, A_n) + 0 + 0 + \det(A_1, \ldots, A_j , \ldots,
-A_i, \ldots, A_n) \iff \det(A_1, \ldots, A_i, \ldots, A_j, \ldots, A_n)
-= -\det(A_1,\ldots, A_j, \ldots, A_i, \ldots, A_n)
+si ha che 0 =\det(A_1, \ldots, A_i + A_j,
+\ldots, A_j + A_i, \ldots, A_n) = \det(A_1, \ldots, A_i, \ldots, A_j +
+A_i, \ldots, A_n) + \det(A_1, \ldots, A_j, \ldots, A_j + A_i, \ldots,
+A_n) =\det(A_1, \ldots, A_i, \ldots, A_j, \ldots, A_n)+\det(A_1, \ldots,
+A_i, \ldots, A_i, \ldots, A_n) + \det(A_1, \ldots, A_j, \ldots,A_j,
+\ldots, A_n) + \det(A_1, \ldots, A_j, \ldots, A_i, \ldots, A_n) =
+\det(A_1, \ldots, A_i, \ldots, A_j, \ldots, A_n) + 0 + 0 + \det(A_1,
+\ldots, A_j , \ldots, A_i, \ldots, A_n) \iff \det(A_1, \ldots, A_i,
+\ldots, A_j, \ldots, A_n) = -\det(A_1,\ldots, A_j, \ldots, A_i, \ldots,
+A_n)
si noti che la tesi è verificata sia per righe che per colonne, per
definizione di \det
@@ -549,9 +549,9 @@
I_n
allora I_n = A \cdot B = (A \cdot B^1,
\ldots, A \cdot B^n) = (\mathscr{L}_A(B^1), \ldots, \mathscr{L}_A(B^n))
-\iff \left \{ \begin{array}{c} \mathscr{L}_A(B^1) = e_1 \\ \vdots \\
-\mathscr{L}_A(B^n) = e_n \end{array} \right.\iff e_1, \ldots, e_n \in
-\textrm{im}(\mathscr{L}_A) \implies \textrm{span}(e_1, \ldots, e_n)
+\iff \left \{ \begin{array}{c} \mathscr{L}_A(B^1) = e_1 \\ \vdots
+\\ \mathscr{L}_A(B^n) = e_n \end{array} \right.\iff e_1, \ldots, e_n
+\in \textrm{im}(\mathscr{L}_A) \implies \textrm{span}(e_1, \ldots, e_n)
\subseteq \textrm{im}(\mathscr{L}_A)
e_1, \ldots, e_n base canonica di
\mathbb{K}^n \implies \dim(\textrm{span}(e_1,
@@ -889,8 +889,8 @@
I_n})
allora, per il teorema del rango \textrm{rk}(\mathscr{L}_{A - \lambda \cdot I_n}) = n
-- \dim(\ker(\mathscr{L}_{A - \lambda \cdot I_n})) \iff
-\dim(\ker(\mathscr{L}_{A - \lambda \cdot I_n})) = n -
+- \dim(\ker(\mathscr{L}_{A - \lambda \cdot I_n}))
+\iff \dim(\ker(\mathscr{L}_{A - \lambda \cdot I_n})) = n -
\textrm{rk}(\mathscr{L}_{A - \lambda \cdot I_n}) =
\dim(\textrm{E}_\lambda(A)) =: \nu(\lambda)
diff --git a/html/everything.html b/html/everything.html
index ea6a2f4..360077b 100644
--- a/html/everything.html
+++ b/html/everything.html
@@ -984,7 +984,7 @@
Th
-- \exists d \in I \mid I = I(d), o
+
- \exists d \in I \mid I = I(d), o
equivalentemente, in \mathbb{Z} ogni
ideale è principale
@@ -997,7 +997,7 @@
- a_{1}, \ldots , a_{n} \in
\mathbb{Z}
-- \exists !d \in \mathbb{N} \mid
+
- \exists !d \in \mathbb{N} \mid
I\left(a_{1}, \ldots , a_{n}\right)=I(d), ed è detto
massimo comun divisore degli a_1,
\ldots, a_n
@@ -1113,7 +1113,7 @@
- a_{1}, \ldots, a_{n} \in
\mathbb{Z}
-- \displaystyle \exists ! m \in \mathbb{N}
+
- \displaystyle \exists ! m \in \mathbb{N}
\mid I(m) = I(a_1) \cap \ldots \cap I(a_n) =
\bigcap_{i=1}^{n}{I(a_i)}, ed è detto minimo comune
multiplo degli a_1, \ldots,
@@ -1178,7 +1178,7 @@
commutativo
- I, J \subset A ideali
- I \cdot J = \{i_1 j_1 + \ldots + i_k j_k
-\mid k \ge 1, i_1 , \ldots , i_k \in I, j_1 , \ldots , j_k \in J
+\mid k \ge 1, i_1 , \ldots , i_k \in I, j_1 , \ldots , j_k \in J
\} è detto prodotto tra I e J
@@ -1376,7 +1376,7 @@
- Th
-- x \equiv y \ (\bmod \ d)
+- x \equiv y \ (\bmod \ d)
Teorema 43
@@ -1993,7 +1993,7 @@
\textrm{sgn}(\sigma) :=
(-1)^{|\textrm{Inv}(\sigma)|} =\left\{\begin{array}{ll}+1 &
-|\operatorname{Inv}(\sigma)| \equiv 0 \ (\bmod \ 2) \\ -1 &
+|\operatorname{Inv}(\sigma)| \equiv 0 \ (\bmod \ 2) \\ -1 &
|\operatorname{Inv}(\sigma)| \equiv 1 \ (\bmod \
2)\end{array}\right.
Th
-- \exists p(x) \in I \mid I =
+
- \exists p(x) \in I \mid I =
I(p(x)), o equivalentemente, in \mathbb{K}[x] ogni ideale è principale
@@ -4711,7 +4711,7 @@
V(b_0, \ldots, b_n) := \left (
\begin{array}{cccc} b_0^0 & b_0^1 & \cdots & b_0^n \\ b_1^0
& b_1^1 & \cdots & b_1^n \\ \vdots & \ddots & &
-\vdots \\\vdots & &\ddots & \vdots \\ b_n^0 & b_n^1
+\vdots \\\vdots & &\ddots & \vdots \\ b_n^0 & b_n^1
& \cdots & b_n^n\end{array}\right) è detta
matrice di Vandermonde a coefficienti b_0, \ldots, b_n
@@ -4748,7 +4748,7 @@
\rightarrow W:(v_1, \ldots, v_n) \rightarrow w
f è detta
multilineare \iff \forall i
-\in [1, n], v_1 , \ldots, v_n \in V_1 \times \ldots \times V_n, v_i,
+\in [1, n], v_1 , \ldots, v_n \in V_1 \times \ldots \times V_n, v_i,
v_i' \in V_i, \lambda, \mu \in \mathbb{K} \quad f(v_1, \ldots,
\lambda v_i+\mu v_i', \ldots, v_n) = \lambda f(v_1, \ldots, v_i,
\ldots, v_n) + \mu f(v_1, \ldots, v_i', \ldots, v_n)
@@ -5532,7 +5532,7 @@
Th
-- z^{n}=|z|^{n} e^{i \theta n} \quad \arg
+
- z^{n}=|z|^{n} e^{i \theta n} \quad \arg
\left( z^{n} \right)=n \arg (z)
@@ -5793,7 +5793,7 @@
Alg
- \forall i \in [0, n] \quad p_i(x) :=
-\displaystyle \prod_{\begin{subarray}{c}0 \le j \le n \\ i \neq\ j
+\displaystyle \prod_{\begin{subarray}{c}0 \le j \le n \\ i \neq\ j
\end{subarray}}{\dfrac{x - b_j}{b_i - b_j}}
- p(x) := c_0p_0(x) + \ldots + c_n
p_n(x)
@@ -5940,9 +5940,9 @@
- Hp
-- a_1, \ldots, a_n \ge 2 \in \mathbb{Z} \mid
-\textrm{MCD}(a_i, a_j) = 1 \quad \forall i, j \in [1, n] : i \neq
-j
+- a_1, \ldots, a_n \ge 2 \in
+\mathbb{Z} \mid \textrm{MCD}(a_i, a_j) = 1 \quad \forall i, j \in [1,
+n] : i \neq j
- m := \textrm{mcm}(a_1, \ldots,
a_n)
@@ -5986,7 +5986,7 @@
- \exists ! x \ (\bmod \ m) \mid
\left\{\begin{array}{c}x \equiv b_{1}\
-\left(\bmod \ a_{1}\right) \\ \vdots \\ x \equiv b_{n}\ \left(\bmod \
+\left(\bmod \ a_{1}\right) \\ \vdots \\ x \equiv b_{n}\ \left(\bmod \
a_{n}\right)\end{array}\right.
diff --git a/html/gruppi-e-anelli.html b/html/gruppi-e-anelli.html
index 1d9041c..a1a094a 100644
--- a/html/gruppi-e-anelli.html
+++ b/html/gruppi-e-anelli.html
@@ -929,7 +929,7 @@
- 0 \in \mathbb{Z} \land g^{0}=e \implies 0
\in I(g)
-- m, n \in \mathbb{Z} \mid g^{m}=g^{n}=e
+
- m, n \in \mathbb{Z} \mid g^{m}=g^{n}=e
\implies g^{m} \cdot g^{n}= g^{m + n} \iff e \cdot e = e \implies m +n
\in I(g) per definizoine di I(g), quindi I(g) +
@@ -1033,7 +1033,7 @@
- Dim
- I(d) = I(g), allora d \in I(d) \implies d \in I(g) \implies g^d =
+class="math inline">d \in I(d) \implies d \in I(g) \implies g^d =
e
- d = o(g) = |H(g)| \bigg\vert |G|
per il teorema di Lagrange, e dunque \exists k
diff --git a/html/ideali.html b/html/ideali.html
index 0c2d119..e6fb331 100644
--- a/html/ideali.html
+++ b/html/ideali.html
@@ -358,9 +358,9 @@
- (I(a_1, \ldots, a_n) , +) \leqslant (A,
+)
-- 0 = a_1 \cdot 0 + \ldots + a_n \cdot 0 \in
-I(a_1, \ldots a_n), dunque 0 è
-l’elemento neutro
+- 0 = a_1 \cdot 0 + \ldots + a_n \cdot 0
+\in I(a_1, \ldots a_n), dunque 0
+è l’elemento neutro
- \forall x, y \in I(a_1, \ldots, a_n) \quad
x = a_1b_1 + \ldots +a_nb_n \land y = a_1c_1 + \ldots+ a_nc_n \implies
x+ y = a_1b_1 + \ldots + a_nb_n + a_1c_1 + \ldots +a_nc_n, che è
@@ -399,7 +399,7 @@
- Th
-- \exists d \in I \mid I = I(d), o
+
- \exists d \in I \mid I = I(d), o
equivalentemente, in \mathbb{Z} ogni
ideale è principale
@@ -477,7 +477,7 @@
- a_{1}, \ldots , a_{n} \in
\mathbb{Z}
-- \exists !d \in \mathbb{N} \mid
+
- \exists !d \in \mathbb{N} \mid
I\left(a_{1}, \ldots , a_{n}\right)=I(d), ed è detto
massimo comun divisore degli a_1,
\ldots, a_n
@@ -528,7 +528,7 @@
- d è il massimo tra i divisori
comuni se \forall k \in \mathbb{Z}: k \mid
-a_1, \ldots, a_n \quad k \mid d
+a_1, \ldots, a_n \quad k \mid d
- \forall i \in [1, n] \quad k \mid a_i \iff
\exists x_i \in \mathbb{Z} \mid kx_i = a_i
- d \in I(d) = I(a_1, \ldots, a_n) \iff d
@@ -666,7 +666,7 @@
- a_{1}, \ldots, a_{n} \in
\mathbb{Z}
-- \displaystyle \exists ! m \in \mathbb{N}
+
- \displaystyle \exists ! m \in \mathbb{N}
\mid I(m) = I(a_1) \cap \ldots \cap I(a_n) =
\bigcap_{i=1}^{n}{I(a_i)}, ed è detto minimo comune
multiplo degli a_1, \ldots,
@@ -715,7 +715,7 @@
- m è il minimo tra i multipli comuni
se \forall k \in \mathbb{Z} : a_1, \ldots, a_n
-\mid k \quad m \mid k
+\mid k \quad m \mid k
- \forall i \in [1, n] \quad a_i \mid k \iff
\exists x_i \in \mathbb{Z} \mid a_ix_i = k \iff k \in I(a_i),
allora k \in I(a_1) \cap \ldots \cap I(a_n) =
@@ -772,7 +772,7 @@
commutativo
- I, J \subset A ideali
- I \cdot J = \{i_1 j_1 + \ldots + i_k j_k
-\mid k \ge 1, i_1 , \ldots , i_k \in I, j_1 , \ldots , j_k \in J
+\mid k \ge 1, i_1 , \ldots , i_k \in I, j_1 , \ldots , j_k \in J
\} è detto prodotto tra I e J
diff --git a/html/matrici.html b/html/matrici.html
index 9dc4864..021420f 100644
--- a/html/matrici.html
+++ b/html/matrici.html
@@ -396,7 +396,7 @@
\end{array}\right) =\right.\left.\left(\begin{array}{c}a_{1, 1} x_1 + \ldots +
a_{1,n}x_n \\ \vdots \\ a_{m,1}x_1 + \ldots + a_{m,n} x_n
-\end{array}\right) = x_1 \left(\begin{array}{c}a_{1, 1} \\ \vdots \\
+\end{array}\right) = x_1 \left(\begin{array}{c}a_{1, 1} \\ \vdots \\
a_{m, 1}\end{array}\right) + \ldots + x_n \left(\begin{array}{c}a_{1,
n}\\ \vdots \\ a_{m, n}\end{array}\right) = x_1A^1 + \ldots + x_nA^n
\right\}=:\textrm{span}(A^1, \ldots, A^n)
@@ -1292,7 +1292,7 @@
- V(b_0, \ldots, b_n) := \left (
\begin{array}{cccc} b_0^0 & b_0^1 & \cdots & b_0^n \\ b_1^0
& b_1^1 & \cdots & b_1^n \\ \vdots & \ddots & &
-\vdots \\\vdots & &\ddots & \vdots \\ b_n^0 & b_n^1
+\vdots \\\vdots & &\ddots & \vdots \\ b_n^0 & b_n^1
& \cdots & b_n^n\end{array}\right) è detta
matrice di Vandermonde a coefficienti b_0, \ldots, b_n
@@ -1317,24 +1317,24 @@
- \det(V(b_0, \ldots, b_n))=\det\left (
\begin{array}{cccc} b_0^0 & b_0^1 & \cdots & b_0^n \\ b_1^0
& b_1^1 & \cdots & b_1^n \\ \vdots & \ddots & &
-\vdots \\\vdots & &\ddots & \vdots \\ b_n^0 & b_n^1
+\vdots \\\vdots & &\ddots & \vdots \\ b_n^0 & b_n^1
& \cdots & b_n^n\end{array}\right) = \det\left (
\begin{array}{cccc} 1 & b_0 & \cdots & b_0^n \\ 1 & b_1
& \cdots & b_1^n \\ \vdots & \ddots & & \vdots
-\\\vdots & &\ddots & \vdots \\ 1 & b_n & \cdots
+\\\vdots & &\ddots & \vdots \\ 1 & b_n & \cdots
& b_n^n\end{array}\right)
- sottraendo la prima riga a tutte le altre, si ottiene \det\left ( \begin{array}{cccc} 1 & b_0 &
\cdots & b_0^n \\ 0 & b_1 - b_0 & \cdots & b_1^n - b_0^n
-\\ \vdots & \ddots & & \vdots \\\vdots & &\ddots
-& \vdots \\ 0 & b_n - b_0 & \cdots & b_n^n -
-b_0^n\end{array}\right)
+\\ \vdots & \ddots & & \vdots \\\vdots &
+&\ddots & \vdots \\ 0 & b_n - b_0 & \cdots & b_n^n
+- b_0^n\end{array}\right)
- eseguendo lo sviluppo di Laplace sulla prima colonna, si ottiene
1 \cdot \det\left ( \begin{array}{cccc} b_1 -
b_0 & b_1^2 -b_0^2 & \cdots & b_1^n - b_0^n \\ b_2 - b_0
& b_2^2-b_0^2&\cdots & b_2^n - b_0^n \\ \vdots & \ddots
-& &\vdots \\ \vdots & & \ddots & \vdots \\ b_n - b_0
-& b_n^2-b_0^2 & \cdots & b_n^n -
+& &\vdots \\ \vdots & & \ddots & \vdots \\ b_n -
+b_0 & b_n^2-b_0^2 & \cdots & b_n^n -
b_0^n\end{array}\right)
- \forall i \in [1, n] \quad b_i^n - b_0^n =
(b_i - b_0)(b_i^{n - 1} + b_i^{n - 2}b_0 + \ldots + b_ib_0^{n - 2} +
@@ -1343,30 +1343,30 @@
\begin{array}{cccc} (b_1 - b_0) \cdot 1 & (b_1 -b_0)\cdot(b_1 + b_0)
& \cdots & (b_1 - b_0)\cdot(b_1^{n - 1} + \ldots + b_0^{ n - 1})
\\ (b_2 - b_0) \cdot 1 & (b_2 - b_0)\cdot (b_2+b_0)&\cdots &
-(b_2 - b_0)\cdot(b_2^{n -1} + \ldots + b_0^{n - 1}) \\ \vdots &
-\ddots & &\vdots \\ \vdots & & \ddots & \vdots \\
-(b_n - b_0) \cdot 1 & (b_n-b_0)\cdot (b_n + b_0) & \cdots &
-(b_n - b_0) \cdot (b_n^{n -1}+ \ldots + b_0^{n -1})\end{array}\right) =
-(b_n - b_0)\cdot \ldots \cdot(b_1 - b_0) \cdot \det\left (
-\begin{array}{cccc} 1 & b_1 + b_0 & \cdots & b_1^{n - 1} +
-\ldots + b_0^{ n - 1} \\1 & b_2+b_0&\cdots & b_2^{n -1} +
-\ldots + b_0^{n - 1} \\ \vdots & \ddots & &\vdots \\ \vdots
-& & \ddots & \vdots \\ 1 & b_n + b_0 & \cdots &
-b_n^{n -1}+ \ldots + b_0^{n -1}\end{array}\right) per
-multilinearità del determinante
+(b_2 - b_0)\cdot(b_2^{n -1} + \ldots + b_0^{n - 1}) \\ \vdots &
+\ddots & &\vdots \\ \vdots & & \ddots & \vdots
+\\ (b_n - b_0) \cdot 1 & (b_n-b_0)\cdot (b_n + b_0) & \cdots
+& (b_n - b_0) \cdot (b_n^{n -1}+ \ldots + b_0^{n
+-1})\end{array}\right) = (b_n - b_0)\cdot \ldots \cdot(b_1 - b_0) \cdot
+\det\left ( \begin{array}{cccc} 1 & b_1 + b_0 & \cdots &
+b_1^{n - 1} + \ldots + b_0^{ n - 1} \\1 & b_2+b_0&\cdots &
+b_2^{n -1} + \ldots + b_0^{n - 1} \\ \vdots & \ddots &
+&\vdots \\ \vdots & & \ddots & \vdots \\ 1 & b_n
++ b_0 & \cdots & b_n^{n -1}+ \ldots + b_0^{n
+-1}\end{array}\right)
per multilinearità del determinante
- \left( \begin{array}{cccc} 1 & b_1 +
b_0 & \cdots & b_1^{n - 1} + \ldots + b_0^{ n - 1} \\1 &
-b_2+b_0&\cdots & b_2^{n -1} + \ldots + b_0^{n - 1} \\ \vdots
+b_2+b_0&\cdots & b_2^{n -1} + \ldots + b_0^{n - 1} \\ \vdots
& \ddots & &\vdots \\ \vdots & & \ddots & \vdots
-\\ 1 & b_n + b_0 & \cdots & b_n^{n -1}+ \ldots + b_0^{n
+\\ 1 & b_n + b_0 & \cdots & b_n^{n -1}+ \ldots + b_0^{n
-1}\end{array}\right)\xrightarrow{\left . \begin{subarray}{c}C^2 -= b_0
-\cdot C^1 \\ C^3 -= b_0 \cdot C^1 + b_0 \cdot C^2 \\ \vdots \\ C^n -=
+\cdot C^1 \\ C^3 -= b_0 \cdot C^1 + b_0 \cdot C^2 \\ \vdots \\ C^n -=
b_0 \cdot C^1 + \ldots + b_0 \cdot C^{n - 1}\end{subarray}\right.} \left
( \begin{array}{cccc} 1 & b_1 & \cdots & b_1^{n - 1} \\1
& b_2 &\cdots & b_2^{n -1} \\ \vdots & \ddots &
-&\vdots \\ \vdots & & \ddots & \vdots \\ 1 & b_n
-& \cdots & b_n^{n -1}\end{array}\right) = V(b_1, \ldots,
-b_n)
+&\vdots \\ \vdots & & \ddots & \vdots \\ 1
+& b_n & \cdots & b_n^{n -1}\end{array}\right) = V(b_1,
+\ldots, b_n)
- allora si ha che \det(V(b_0, \ldots, b_n))
= (b_n - b_0) \cdot \ldots \cdot(b_1 - b_0) \cdot \det(V(b_1, \ldots,
b_n)) = (b_n - b_0) \cdot \ldots \cdot(b_1 - b_0) \cdot (b_n - b_1)
diff --git a/html/numeri-complessi.html b/html/numeri-complessi.html
index b6611e0..3d845be 100644
--- a/html/numeri-complessi.html
+++ b/html/numeri-complessi.html
@@ -362,7 +362,7 @@
Th
-- z^{n}=|z|^{n} e^{i \theta n} \quad \arg
+
- z^{n}=|z|^{n} e^{i \theta n} \quad \arg
\left( z^{n} \right)=n \arg (z)
diff --git a/html/permutazioni.html b/html/permutazioni.html
index fc41dff..f0f9a67 100644
--- a/html/permutazioni.html
+++ b/html/permutazioni.html
@@ -530,7 +530,7 @@
\textrm{sgn}(\sigma) :=
(-1)^{|\textrm{Inv}(\sigma)|} =\left\{\begin{array}{ll}+1 &
-|\operatorname{Inv}(\sigma)| \equiv 0 \ (\bmod \ 2) \\ -1 &
+|\operatorname{Inv}(\sigma)| \equiv 0 \ (\bmod \ 2) \\ -1 &
|\operatorname{Inv}(\sigma)| \equiv 1 \ (\bmod \
2)\end{array}\right.
@@ -580,18 +580,18 @@
- Dim
- \tau_{i, i +
-1}=\left(\begin{array}{ccccccc}1 & \cdots & i & & i + 1
-& \cdots & n \\ 1 & \cdots & i + 1 & & i &
+1}=\left(\begin{array}{ccccccc}1 & \cdots & i & & i + 1
+& \cdots & n \\ 1 & \cdots & i + 1 & & i &
\cdots & n\end{array}\right) è una trasposizione
adiacente
- \sigma=\left(\begin{array}{ccccccc}1 &
-\cdots & i & & i + 1 & \cdots & n \\ \sigma(1) &
-\cdots & \sigma(i) & & \sigma(i + 1) & \cdots &
-\sigma(n) \end{array}\right) è una permutazione
+\cdots & i & & i + 1 & \cdots & n \\ \sigma(1)
+& \cdots & \sigma(i) & & \sigma(i + 1) & \cdots
+& \sigma(n) \end{array}\right) è una permutazione
- allora \sigma \tau_{i, i +
-1}=\left(\begin{array}{ccccccc}1 & \cdots & i & & i + 1
-& \cdots & n \\ \sigma(1) & \cdots & \sigma(i + 1) &
-& \sigma(i) & \cdots & \sigma(n)
+1}=\left(\begin{array}{ccccccc}1 & \cdots & i & & i + 1
+& \cdots & n \\ \sigma(1) & \cdots & \sigma(i + 1)
+& & \sigma(i) & \cdots & \sigma(n)
\end{array}\right)
- \forall i \in [1, n] \quad i \lt i + 1
\implies \sigma \tau_{i, i + 1}(i + 1) \gt \sigma \tau_{i, i + 1}(i)
@@ -622,7 +622,7 @@
- allora (-1)^k \cdot \textrm{sgn}(\sigma) =
\textrm{sgn}(\textrm{id}) = 1
- \textrm{sgn}(\sigma) = \pm 1, e
-poiché (-1)^k \cdot \textrm{sgn}(\sigma) =
+poiché (-1)^k \cdot \textrm{sgn}(\sigma) =
1, allora necessariamente \textrm{sgn}(\sigma) = (-1)^k
@@ -781,10 +781,10 @@
esempio
- \begin{aligned} \sigma\ =\
-&(13)(254)(876) \\ & \ \downarrow \downarrow \ \ \ \downarrow
-\downarrow \downarrow \ \ \ \downarrow \downarrow \downarrow \\
-\sigma^{\prime}=\ &(25)(184)(376) \end{aligned} \implies \implies\alpha=\left(\begin{array}{llllllll}1 & 2 &
3 & 4 & 5 & 6 & 7 & 8 \\ 2 & 1 & 5 & 4
& 8 & 6 & 7 & 3\end{array}\right)
@@ -928,7 +928,7 @@
& 2 & \cdots & d_{1} & d_{1}+1 & \cdots &
d_{1}+d_{2} & \cdots & \cdots & n - d_k + 1 & \cdots
& n \\ 2 & 3 & \cdots & 1 & d_{1}+2 & \cdots
-& d_{1}+1 & \cdots & \cdots & n - d_k + 2 & \cdots
+& d_{1}+1 & \cdots & \cdots & n - d_k + 2 & \cdots
& n - d_k + 1\end{array}\right)
- si noti che, ad esempio, per portare
class="math inline">\cdot sono ben definite
- p(x), q(x) \in \mathbb{K}[x] \mid \left \{
-\begin{array}{l} p(x) = \displaystyle{\sum_{i = 0}^n a_ix^i} \\ q(x) =
+\begin{array}{l} p(x) = \displaystyle{\sum_{i = 0}^n a_ix^i} \\ q(x) =
\displaystyle{\sum_{j = 0}^m b_jx^j} \end{array} \right.
- allora p(x) + q(x) = \displaystyle{\sum_{k
= 0}^{+\infty}{(a_k + b_k)}}x^k
@@ -498,7 +498,7 @@
- Th
-- \exists p(x) \in I \mid I =
+
- \exists p(x) \in I \mid I =
I(p(x)), o equivalentemente, in \mathbb{K}[x] ogni ideale è principale
diff --git a/html/relazioni.html b/html/relazioni.html
index 7203e4c..1e6eda6 100644
--- a/html/relazioni.html
+++ b/html/relazioni.html
@@ -324,8 +324,8 @@
- riflessività
-- x\mid x \iff \exists p \in \mathbb{N} \mid
-x p=x \iff p = 1 \in \mathbb{N}
+- x\mid x \iff \exists p \in \mathbb{N}
+\mid x p=x \iff p = 1 \in \mathbb{N}
- transitività
- Th
-- x \equiv y \ (\bmod \ d)
+- x \equiv y \ (\bmod \ d)
- Dim
-- x \equiv y \ (\bmod \ n) \iff n \mid y - x
-\iff \exists p \in \mathbb{Z} \mid np = y - x
+- x \equiv y \ (\bmod \ n) \iff n \mid y -
+x \iff \exists p \in \mathbb{Z} \mid np = y - x
- d \mid n \iff \exists k \in \mathbb{Z}
\mid dk = n
- allora, np = y - x \iff dkp = y -x
diff --git a/html/spazi-vettoriali.html b/html/spazi-vettoriali.html
index ab4734f..0e7d99c 100644
--- a/html/spazi-vettoriali.html
+++ b/html/spazi-vettoriali.html
@@ -1049,7 +1049,7 @@
\begin{array}{l}u \in U \implies u := \displaystyle \sum_{i =
1}^k{\lambda_iw_i} + \sum_{j = k + 1}^m{\lambda_j u_j} \\ v \in V
\implies \displaystyle v :=\sum_{i = 1}^k{\mu_iw_i} + \sum_{h = k + 1}^n
-{\mu_hv_h} \end{array} \right.\iff \displaystyle w := u + v =\sum_{i =
+{\mu_hv_h} \end{array} \right.\iff \displaystyle w := u + v =\sum_{i =
1}^k{(\lambda_i + \mu_i)w_i} + \sum_{j = k + 1}^m{\lambda_j u_j} +
\sum_{h = k + 1}^n {\mu_hv_h} \iff w := u + v \in
\textrm{span}(\mathcal{B}_1 \cup \mathcal{B}_2) \implies U + V \subseteq
@@ -1063,7 +1063,7 @@
- \mathcal{B}_1 \cup \mathcal{B}_2
generatori di U + V \implies \exists
\lambda_1, \ldots, \lambda_k, \mu_{k + 1}, \ldots, \mu_m, \eta_{k + 1},
-\ldots, \eta_n \in \mathbb{K} \mid \displaystyle \sum_{i =
+\ldots, \eta_n \in \mathbb{K} \mid \displaystyle \sum_{i =
1}^k{\lambda_iw_i} +\sum_{j = k +1}^m{\mu_j u_j } +\displaystyle \sum_{h
= k + 1} ^n{\eta_hv_h} = 0_W \in U + V
- siano \left \{ \begin{array}{l}a :=
diff --git a/html/teoremi-fondamentali.html b/html/teoremi-fondamentali.html
index 4cc1cb9..e0441c1 100644
--- a/html/teoremi-fondamentali.html
+++ b/html/teoremi-fondamentali.html
@@ -281,7 +281,7 @@
- unicità
- per assurdo \exists q_1(x), q_2(x),
-r_1(x), r_2(x) \in \mathbb{K}[x] \mid \left \{ \begin{array}{l} a(x) =
+r_1(x), r_2(x) \in \mathbb{K}[x] \mid \left \{ \begin{array}{l} a(x) =
b(x)q_1(x) + r_1(x) & \deg(r_1(x)) \lt \deg(b(x)) \\ a(x) =
b(x)q_2(x) + r_2(x) & \deg(r_2(x)) \lt \deg(b(x))\end{array}
\right.
@@ -344,8 +344,8 @@
\mathbb{P} : n = 2^{n_2} \cdot 3 ^ {n_3} \cdot \ldots \cdot p ^
{n_p}
-- p \nmid n \implies n_p = 0 \implies p ^
-{n_p} = 1, dunque non influisce nella produttoria
+- p \nmid n \implies n_p = 0 \implies p
+^ {n_p} = 1, dunque non influisce nella produttoria
- \displaystyle{n = \prod_{p \in
\mathbb{P}}^{} p ^{n_p}}, quindi è possibile riscrivere anche
@@ -390,9 +390,9 @@
- Hp
-- a_1, \ldots, a_n \ge 2 \in \mathbb{Z} \mid
-\textrm{MCD}(a_i, a_j) = 1 \quad \forall i, j \in [1, n] : i \neq
-j
+- a_1, \ldots, a_n \ge 2 \in
+\mathbb{Z} \mid \textrm{MCD}(a_i, a_j) = 1 \quad \forall i, j \in [1,
+n] : i \neq j
- m := \textrm{mcm}(a_1, \ldots,
a_n)
@@ -486,7 +486,7 @@
- \exists ! x \ (\bmod \ m) \mid
\left\{\begin{array}{c}x \equiv b_{1}\
-\left(\bmod \ a_{1}\right) \\ \vdots \\ x \equiv b_{n}\ \left(\bmod \
+\left(\bmod \ a_{1}\right) \\ \vdots \\ x \equiv b_{n}\ \left(\bmod \
a_{n}\right)\end{array}\right.
- Dim
@@ -575,8 +575,8 @@
[1] in \mathbb{Z}_{n_1}^*, e
dunque si ottiene che \left\{\begin{array}{c}a^m \equiv 1 \ (\bmod \ n_1)
-\\ \vdots \\ a^m \equiv 1 \ (\bmod \ n_k)\end{array}\right. \implies a^m
-\equiv 1 \ (\bmod \ N) \implies m è multiplo di è multiplo di o \implies o \mid m
- m \mid o \land o \mid m \implies o =
m
@@ -692,7 +692,7 @@
- Dim
- per il piccolo teorema di Fermat a^p
-\equiv a \iff a^{p - 1} \equiv 1 \iff a^{p - 1} - 1 \equiv 0\ (\bmod \
+\equiv a \iff a^{p - 1} \equiv 1 \iff a^{p - 1} - 1 \equiv 0\ (\bmod \
p), ovvero [a] è radice del
polinomio x^{p -1 } - [1] \in
\mathbb{Z}_p[x]
diff --git a/mds/determinante.md b/mds/determinante.md
index 906b313..3b56ac5 100644
--- a/mds/determinante.md
+++ b/mds/determinante.md
@@ -50,7 +50,7 @@
- $\det(A_1, \ldots, A_i, \ldots, A_j , \ldots, A_n) = -\det(A_1, \ldots, A_j, \ldots, A_i, \ldots, A_n)$
- **Dim**
- per il punto 2 della definizione di $\det$ si ha che $\det(A_1, \ldots, A_i + A_j, \ldots, A_j + A_i, \ldots, A_n) = 0$
- - allora, per multilinearità di $\det$ si ha che $\det$ si ha che $0 =\det(A_1, \ldots, A_i + A_j, \ldots, A_j + A_i, \ldots, A_n) = \det(A_1, \ldots, A_i, \ldots, A_j + A_i, \ldots, A_n) + \det(A_1, \ldots, A_j, \ldots, A_j + A_i, \ldots, A_n) =\det(A_1, \ldots, A_i, \ldots, A_j, \ldots, A_n)+\det(A_1, \ldots, A_i, \ldots, A_i, \ldots, A_n) + \det(A_1, \ldots, A_j, \ldots,A_j, \ldots, A_n) + \det(A_1, \ldots, A_j, \ldots, A_i, \ldots, A_n) = \det(A_1, \ldots, A_i, \ldots, A_j, \ldots, A_n) + 0 + 0 + \det(A_1, \ldots, A_j , \ldots, A_i, \ldots, A_n) \iff \det(A_1, \ldots, A_i, \ldots, A_j, \ldots, A_n) = -\det(A_1,\ldots, A_j, \ldots, A_i, \ldots, A_n)$
+ - allora, per multilinearità di $\det$ si ha che $0 =\det(A_1, \ldots, A_i + A_j, \ldots, A_j + A_i, \ldots, A_n) = \det(A_1, \ldots, A_i, \ldots, A_j + A_i, \ldots, A_n) + \det(A_1, \ldots, A_j, \ldots, A_j + A_i, \ldots, A_n) =\det(A_1, \ldots, A_i, \ldots, A_j, \ldots, A_n)+\det(A_1, \ldots, A_i, \ldots, A_i, \ldots, A_n) + \det(A_1, \ldots, A_j, \ldots,A_j, \ldots, A_n) + \det(A_1, \ldots, A_j, \ldots, A_i, \ldots, A_n) = \det(A_1, \ldots, A_i, \ldots, A_j, \ldots, A_n) + 0 + 0 + \det(A_1, \ldots, A_j , \ldots, A_i, \ldots, A_n) \iff \det(A_1, \ldots, A_i, \ldots, A_j, \ldots, A_n) = -\det(A_1,\ldots, A_j, \ldots, A_i, \ldots, A_n)$
- si noti che la tesi è verificata sia per righe che per colonne, per definizione di $\det$
## Oss