From 2da1fb606a49c3ad690377a32c47d222acbb0602 Mon Sep 17 00:00:00 2001 From: GitHub Action Date: Thu, 26 Oct 2023 12:53:20 +0000 Subject: [PATCH] Add changes for 758b8b98c63e125d32424ae5e35d633cc6a8f02d --- .doctrees/assets/index.doctree | Bin 77938 -> 92657 bytes .../notes/algebra/algebra_2023.10.17.doctree | Bin 21338 -> 19454 bytes .../notes/algebra/algebra_2023.10.24.doctree | Bin 0 -> 14243 bytes .../matematyka/matematyka_2023.10.15.doctree | Bin 12627 -> 12645 bytes .../matematyka/matematyka_2023.10.24.doctree | Bin 0 -> 5230 bytes .doctrees/environment.pickle | Bin 262012 -> 309255 bytes assets/index.html | 142 +++++++++++- assets/notes/algebra/algebra_2023.10.17.html | 17 +- assets/notes/algebra/algebra_2023.10.24.html | 215 ++++++++++++++++++ .../matematyka/matematyka_2023.10.15.html | 2 +- .../matematyka/matematyka_2023.10.24.html | 174 ++++++++++++++ index.html | 1 + objects.inv | Bin 497 -> 545 bytes searchindex.js | 2 +- 14 files changed, 530 insertions(+), 23 deletions(-) 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Własności działań -(1)\[\begin{bmatrix} +
+(1)\[\begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix}\]
@@ -354,19 +354,88 @@

Macierz odwrotnaInformacja

Macierz jest odwracalna jeżeli jest niosobliwa

+
+

Notatki z pliku notes/notes/algebra/algebra_2023.10.24.md

+
+

Ważne

+

Macierze odwracalne tworzą grupę (tzw. grupę liniową)

+

Obliczanie macierzy odwrotnej

    -

  • Macierz odwrotna to taka macierz, która składa się z odpowiednich dopełnień arytmetycznych -wszystkich elementów macierzy początkowej.

    • -

  • dopełnienie arytmetyczne elementu a_ij to wyznacznik macierzyz z wykreślon itym rzędem i jotą kolumną pomnożony przez \(\fra{1}{detA}\)

    • +

  • metoda dopełnień algebaicznych

    +
      +

    • możemy ją obliczyć tylko gdy \(detA \neq 0\)

      • +

    • \(a_ij = (-1)^{i+j} * m_{ij}\)

      • +

    • tak powstałą macierz transponujemy i kążdy element mnożymy przez \(\frac{1}{detA}\)

      • +
    +
    • +

  • metoda operacji elementarnych (bezwyznacznikiowa)

    +
      +

    • zestawienie macierzy blokowej z macierzy A i macierzy jednostkowej

      • +

    • jeżeli macierz A uda się sprowadzić do macierzy jednostkowej, to druga część bloku +będzie macierzą odwrotną \(A^{-1}\)

      • +
    +
    • +
+
+

Algorytm gausa

+
    +

  • korzystając z operacji elementarnych uzyskać macierz trujkątną górną +przemieszczając się od lewego górnego rogu macierzy

    • +

  • zerowanie od prawej na lewą stronę

    • +
+
+
+
+

Własności macierzy odwrotnej

+
    +

  • \(detA = \frac{1}{detA^{-1}}\)

    • +

  • \((A^{-1})^{-1} = A\)

    • +

  • \((A^{-1})^T = (A^T)^{-1}\)

    • +

  • \((\alpha A)^{-1} = \frac{1}{\alpha} A^{-1}\)

    • +

  • (AB)^-1 = B^{-1} A^{-1}$

    • +
+
+ + +
+

Układy równań liniowych

+

Układ równań liniowych można zapisać jako \(A * X = B\)

+
+

Układ równań

+
+

układy crammerowskie

+

są to takie układy, w których liczba niewiadomych jest równa liczbie równań

+
+
    +

  • Macierz A jest kwadratowa

    • +

  • \(x = \frac{W}{W_x}\) …

    • +

  • jeśli W = 0 układ nie jest oznaczony

    • +

  • jeśli macierz nie jest kwadratowa należy wykorzystać rząd macierzy

    +
      +

    • rząd to największy stopień niezerowego minora

      • +
    +
    • +
+
+
+

Wyznaczanie rzędu

+
    +

  • sprowadzanie macierzy do postaci schodkowej

    • +

  • rząd macierzy to ilość “schodów”

    • +
+
+
+

Rozwiązywanie układów równań \(m \neq n\)

+
    +

  • układ jest sprzeczny jeżei rząd macierzy jest rożny od rzędu macierzy uzupełnionej

Notatki z pliku notes/notes/matematyka_cw/matematyka_cw_2023.10.19.md

-

Matematyka - ćwiczenia

@@ -473,7 +542,7 @@

ważne granice \[\begin{split} -\lim_{h \to 0} \frac{a^h - 1}{h} \\ +\lim_{h \to 0} \frac{a^h - 1}{h} = logn~a \\ \end{split}\]

@@ -577,6 +646,57 @@

Przykłady funkcji ciągłych\(\root{n}{x}\) (funkcja odwrotna do \(x^n\))

  • \(f(x) = a^x~a > 0 \land a \neq 1\)

    • +
    +

    Notatki z pliku notes/notes/matematyka/matematyka_2023.10.24.md

    +
    +\[ +cosh (x) = \frac{e^x + e^{-x}}{2} +sinh (x) = \frac{e^x - e^{-x}}{2} +tgh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} +\]
    + +
    +

    Interpretacja krzywej

    +

    Opis parametryczny krzywej:

    +
      +

    • Okrąg

      • +
    +
    +\[\begin{split} +\left\{\begin{matrix} +x = r sin(t) \\ +y = r cos(t) +\end{matrix}\right. +\end{split}\]
    +
      +

    • Elipsa

      • +
    +
    +\[\begin{split} +\left\{\begin{matrix} +x = a cos(t) \\ +y = b sin(t) +\end{matrix}\right. +\end{split}\]
    +
      +

    • Hiperbola

      • +
    +
    +\[\begin{split} +\left\{\begin{matrix} +x = a cosh(t) \\ +y = b sinh(t) +\end{matrix}\right. +\end{split}\]
    +
    +

    Informacja

    +
    +\[ +arsinh (x) = logn \left(x + \sqrt{1+x}\right) +\]
    +
    +
    +

    Notatki z pliku notes/notes/matematyka/.matematyka_2023.10.24.md.swp

    @@ -616,8 +736,15 @@

    Spis treści:

  • Macierz odwrotna +
    • +
  • Układy równań liniowych
    • @@ -640,6 +767,7 @@

    Spis treści:

  • Funkcje ciągłe
    • diff --git a/assets/notes/algebra/algebra_2023.10.17.html b/assets/notes/algebra/algebra_2023.10.17.html index 7239cdc3..979226da 100644 --- a/assets/notes/algebra/algebra_2023.10.17.html +++ b/assets/notes/algebra/algebra_2023.10.17.html @@ -108,8 +108,8 @@

    Własności działań -(1)\[\begin{bmatrix} +
    +(1)\[\begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix}\]
    @@ -191,14 +191,6 @@

    Macierz odwrotnaInformacja

    Macierz jest odwracalna jeżeli jest niosobliwa

    -
    -

    Obliczanie macierzy odwrotnej

    -
      -

    • Macierz odwrotna to taka macierz, która składa się z odpowiednich dopełnień arytmetycznych -wszystkich elementów macierzy początkowej.

      • -

    • dopełnienie arytmetyczne elementu a_ij to wyznacznik macierzyz z wykreślon itym rzędem i jotą kolumną pomnożony przez \(\fra{1}{detA}\)

      • -
    -
    @@ -220,10 +212,7 @@

    Spis treści:

  • zmiany na macierzy nie zmieniające wyznacznika
    • -
  • Macierz odwrotna -
    • +
  • Macierz odwrotna
    • diff --git a/assets/notes/algebra/algebra_2023.10.24.html b/assets/notes/algebra/algebra_2023.10.24.html new file mode 100644 index 00000000..13c75a78 --- /dev/null +++ b/assets/notes/algebra/algebra_2023.10.24.html @@ -0,0 +1,215 @@ + + + + + + + + + + Obliczanie macierzy odwrotnej — Matematyka - dokumentacja + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + + +
    + + +
    +
    +
    +
    + +
    +

    Ważne

    +

    Macierze odwracalne tworzą grupę (tzw. grupę liniową)

    +
    +
    +

    Obliczanie macierzy odwrotnej

    +
      +

    • metoda dopełnień algebaicznych

      +
        +

      • możemy ją obliczyć tylko gdy \(detA \neq 0\)

        • +

      • \(a_ij = (-1)^{i+j} * m_{ij}\)

        • +

      • tak powstałą macierz transponujemy i kążdy element mnożymy przez \(\frac{1}{detA}\)

        • +
      +
      • +

    • metoda operacji elementarnych (bezwyznacznikiowa)

      +
        +

      • zestawienie macierzy blokowej z macierzy A i macierzy jednostkowej

        • +

      • jeżeli macierz A uda się sprowadzić do macierzy jednostkowej, to druga część bloku +będzie macierzą odwrotną \(A^{-1}\)

        • +
      +
      • +
    +
    +

    Algorytm gausa

    +
      +

    • korzystając z operacji elementarnych uzyskać macierz trujkątną górną +przemieszczając się od lewego górnego rogu macierzy

      • +

    • zerowanie od prawej na lewą stronę

      • +
    +
    +
    +
    +

    Własności macierzy odwrotnej

    +
      +

    • \(detA = \frac{1}{detA^{-1}}\)

      • +

    • \((A^{-1})^{-1} = A\)

      • +

    • \((A^{-1})^T = (A^T)^{-1}\)

      • +

    • \((\alpha A)^{-1} = \frac{1}{\alpha} A^{-1}\)

      • +

    • (AB)^-1 = B^{-1} A^{-1}$

      • +
    +
    +
    +

    Układy równań liniowych

    +

    Układ równań liniowych można zapisać jako \(A * X = B\)

    +
    +

    Układ równań

    +
    +

    układy crammerowskie

    +

    są to takie układy, w których liczba niewiadomych jest równa liczbie równań

    +
    +
      +

    • Macierz A jest kwadratowa

      • +

    • \(x = \frac{W}{W_x}\) …

      • +

    • jeśli W = 0 układ nie jest oznaczony

      • +

    • jeśli macierz nie jest kwadratowa należy wykorzystać rząd macierzy

      +
        +

      • rząd to największy stopień niezerowego minora

        • +
      +
      • +
    +
    +
    +

    Wyznaczanie rzędu

    +
      +

    • sprowadzanie macierzy do postaci schodkowej

      • +

    • rząd macierzy to ilość “schodów”

      • +
    +
    +
    +

    Rozwiązywanie układów równań \(m \neq n\)

    +
      +

    • układ jest sprzeczny jeżei rząd macierzy jest rożny od rzędu macierzy uzupełnionej

      • +
    +
    +
    + + +
    +
    +
    +
    + + + +
    + + + +
    + +
    +
    +
    + + +
    + + + + + \ No newline at end of file diff --git a/assets/notes/matematyka/matematyka_2023.10.15.html b/assets/notes/matematyka/matematyka_2023.10.15.html index f024d884..ec2d1f25 100644 --- a/assets/notes/matematyka/matematyka_2023.10.15.html +++ b/assets/notes/matematyka/matematyka_2023.10.15.html @@ -94,7 +94,7 @@

    ważne granice \[\begin{split} -\lim_{h \to 0} \frac{a^h - 1}{h} \\ +\lim_{h \to 0} \frac{a^h - 1}{h} = logn~a \\ \end{split}\] diff --git a/assets/notes/matematyka/matematyka_2023.10.24.html b/assets/notes/matematyka/matematyka_2023.10.24.html new file mode 100644 index 00000000..322ca2a1 --- /dev/null +++ b/assets/notes/matematyka/matematyka_2023.10.24.html @@ -0,0 +1,174 @@ + + + + + + + + + + Interpretacja krzywej — Matematyka - dokumentacja + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + + +
    + + +
    +
    +
    +
    + +
    +\[ +cosh (x) = \frac{e^x + e^{-x}}{2} +sinh (x) = \frac{e^x - e^{-x}}{2} +tgh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} +\]
    +
    +

    Interpretacja krzywej

    +

    Opis parametryczny krzywej:

    +
      +

    • Okrąg

      • +
    +
    +\[\begin{split} +\left\{\begin{matrix} +x = r sin(t) \\ +y = r cos(t) +\end{matrix}\right. +\end{split}\]
    +
      +

    • Elipsa

      • +
    +
    +\[\begin{split} +\left\{\begin{matrix} +x = a cos(t) \\ +y = b sin(t) +\end{matrix}\right. +\end{split}\]
    +
      +

    • Hiperbola

      • +
    +
    +\[\begin{split} +\left\{\begin{matrix} +x = a cosh(t) \\ +y = b sinh(t) +\end{matrix}\right. +\end{split}\]
    +
    +

    Informacja

    +
    +\[ +arsinh (x) = logn \left(x + \sqrt{1+x}\right) +\]
    +
    +
    + + +
    +
    +
    +
    + + + +
    + + + +
    + +
    +
    +
    + + +
    + + + + + \ No newline at end of file diff --git a/index.html b/index.html index 5472f170..6fed1a33 100644 --- a/index.html +++ b/index.html @@ -80,6 +80,7 @@

    WstępCiało liczb zespolonych
  • Zestaw 2 zadanie 4, przykład a
  • Macierze
    • +
  • Układy równań liniowych
  • Matematyka - ćwiczenia
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