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PÅ‚aszczyzny w przestrzeni \(\mat

Przestrzenie wektorowe¶

Przestrzeń Wektorowa

-

(K, +, \(*\)) jest grupÄ… abelowÄ…, natomiast V zbiorem (\(V \neq \emptyset\)).

+

(K, +, C, \(*\)) jest grupÄ… abelowÄ…, natomiast V zbiorem (\(V \neq \emptyset\)).

  • (V, +) jest grupÄ… abelowÄ…

  • \(\forall v, v \in V \quad \forall \alpha \in K \quad \alpha * (v + v) = (\alpha * v) + (\alpha * v)\) (wystÄ™puje rozdzielność mnożenia wzglÄ™dem dodawania w mnożeniu sumy wektorów przez skalar)

  • @@ -506,6 +506,84 @@

    Przestrzenie wektoroweWskazówka

    Elementy zbioru V nazywamy wektorami, a K - skalarami

+
+

Informacja

+

Zbiór \((\mathbb{R}^2, +, \mathbb{R}, *)\) to tradycyjny zbiór wektorów na płaszczyźnie dwuwymiarowej. +\(V = \mathbb{R}^2\) określa dwa wymiary palszczyzny a \(K = \mathbb{R}\) to ciało (zbiór) tzw. skalarów.

+
+
+

Podprzestrzenie liniowe¶

+

Zbiór \(U\) nazywamy podprzestrzenią przestrzeni V gdy działania zadeklarowane dla te podprzestrzeni są w niej wewnętrzne.

+
+

Informacja

+

Innymi słowy aby sprawdzić czy dany zbiór jest podprzestrzenią przestrzeni V dla dla dwuch wektorów \(v_1 i v_2 \in V\) oraz skalara \(\alpha\)

+
+\[ +v_1 + v_2 \in V' \land \alpha v_1 \in V' +\]
+
+
+

Ważne

+

0 musi należeć do podprzestrzeni

+
+\[\begin{split} +niech~(\mathbb{R^3}, +, \mathbb{R}, *) \\ +-1 * v_1 = -v_1 \in \mathbb{R}^3 \\ +v_1 - v_1 = 0 \\ +\end{split}\]
+
+
+

Wskazówka

+

Jeżeli w definicji przestrzeni występuje translacja (dodanie stałej) przestrzeń ta nie jest podprzestrzenią przestrzeni V.

+
+\[\begin{split} +V = (\mathbb{R}^3, +, K, *) \\ +U = \left{(x + 1, y + x, z) : x, y, z \in \mathbb{R} \right} \\ +\\ +x_1 + 1 + x_2 + 1 = (x_1 + x_2) + 2 \neq (x_1 + x_2) + 1 +\\ +\\ +V = (\mathbb{R}^3, +, K, *) \\ +U = \left{(2x, y + x, z) : x, y, z \in \mathbb{R} \right} \\ +\\ +2x_1 + 2x_2 = 2(x_1 + x_2) \\ +\alpha 2 x_1 + \alpha 2 x_2 = \alpha 2 (x_1 + x_2) +\end{split}\]
+

Analogicznie można postępować dla innych wyrazów.

+
+
+
+

Liniowa zależność wektorów¶

+
+

liniowa zależność wektorów

+

Wektory są liniowo niezależne gdy żadnego z nich nie można przedstawić w formie kombinacji liniowej pozostałych.

+

Przykładowo \(v = \alpha v_1 + \beta v_2 + \gamma v_3\) nie jest liniowo niezależny.

+

Biorąc pod uwagę inny przykład:

+
+\[\begin{split} +v = (5, 4, -18) \\ +v = 5 \hat{i} + 4 \hat{j} - 18 \hat{k} \\ +gdzie:\\ +\hat{i} = (1, 0, 0) \\ +\hat{j} = (0, 1, 0) \\ +\hat{k} = (0, 0, 1) \\ +\end{split}\]
+
+
+

Generatory

+

Zbiór generatorów to zbiór wektorów pozwalających na zapisanie każdego +innego wektora w danej podprzestrzeni.

+
+\[ +v = \alpha w_1 + \beta w_2 + ... + \theta w_n +\]
+
    +
  • zbiór generatorów oznaczamy jako \(W\)

  • +
  • natomiast generowanÄ… przez nie podprzestrzeÅ„ \(limW\)

  • +
+
+
+

Notatki z pliku notes/algebra/.algebra_2023.11.07.md.swp


Notatki z pliku notes/algebra/algebra_2023.11.14.md

podzbiorami przestrzeni wektorowych \(\mathbb{R}^2\) i \(\mathbb{R}^3\) sÄ…:

@@ -544,6 +622,7 @@

Przestrzenie wektorowewektory generują przestrzeń \(\mathbb{R}^n \Leftrightarrow\) rząd macierzy złożonej z tych wektorów jes trówny n

+

Baza¶

Układ wektorów można nazwać bazą, jeżeli:

@@ -1475,6 +1554,8 @@

Spis treści:

  • Przestrzenie wektorowe
      +
    • Podprzestrzenie liniowe
    • +
    • Liniowa zależność wektorów
    • Baza diff --git a/assets/notes/algebra/algebra_2023.11.07.html b/assets/notes/algebra/algebra_2023.11.07.html index 5d8141ea..23ac2bb1 100644 --- a/assets/notes/algebra/algebra_2023.11.07.html +++ b/assets/notes/algebra/algebra_2023.11.07.html @@ -68,7 +68,7 @@

      Nawigacja

      Przestrzenie wektorowe¶

      Przestrzeń Wektorowa

      -

      (K, +, \(*\)) jest grupÄ… abelowÄ…, natomiast V zbiorem (\(V \neq \emptyset\)).

      +

      (K, +, C, \(*\)) jest grupÄ… abelowÄ…, natomiast V zbiorem (\(V \neq \emptyset\)).

      • (V, +) jest grupÄ… abelowÄ…

      • \(\forall v, v \in V \quad \forall \alpha \in K \quad \alpha * (v + v) = (\alpha * v) + (\alpha * v)\) (wystÄ™puje rozdzielność mnożenia wzglÄ™dem dodawania w mnożeniu sumy wektorów przez skalar)

      • @@ -82,6 +82,83 @@

        Przestrzenie wektoroweWskazówka

        Elementy zbioru V nazywamy wektorami, a K - skalarami

      +
      +

      Informacja

      +

      Zbiór \((\mathbb{R}^2, +, \mathbb{R}, *)\) to tradycyjny zbiór wektorów na płaszczyźnie dwuwymiarowej. +\(V = \mathbb{R}^2\) określa dwa wymiary palszczyzny a \(K = \mathbb{R}\) to ciało (zbiór) tzw. skalarów.

      +
      +
      +

      Podprzestrzenie liniowe¶

      +

      Zbiór \(U\) nazywamy podprzestrzenią przestrzeni V gdy działania zadeklarowane dla te podprzestrzeni są w niej wewnętrzne.

      +
      +

      Informacja

      +

      Innymi słowy aby sprawdzić czy dany zbiór jest podprzestrzenią przestrzeni V dla dla dwuch wektorów \(v_1 i v_2 \in V\) oraz skalara \(\alpha\)

      +
      +\[ +v_1 + v_2 \in V' \land \alpha v_1 \in V' +\]
      +
      +
      +

      Ważne

      +

      0 musi należeć do podprzestrzeni

      +
      +\[\begin{split} +niech~(\mathbb{R^3}, +, \mathbb{R}, *) \\ +-1 * v_1 = -v_1 \in \mathbb{R}^3 \\ +v_1 - v_1 = 0 \\ +\end{split}\]
      +
      +
      +

      Wskazówka

      +

      Jeżeli w definicji przestrzeni występuje translacja (dodanie stałej) przestrzeń ta nie jest podprzestrzenią przestrzeni V.

      +
      +\[\begin{split} +V = (\mathbb{R}^3, +, K, *) \\ +U = \left{(x + 1, y + x, z) : x, y, z \in \mathbb{R} \right} \\ +\\ +x_1 + 1 + x_2 + 1 = (x_1 + x_2) + 2 \neq (x_1 + x_2) + 1 +\\ +\\ +V = (\mathbb{R}^3, +, K, *) \\ +U = \left{(2x, y + x, z) : x, y, z \in \mathbb{R} \right} \\ +\\ +2x_1 + 2x_2 = 2(x_1 + x_2) \\ +\alpha 2 x_1 + \alpha 2 x_2 = \alpha 2 (x_1 + x_2) +\end{split}\]
      +

      Analogicznie można postępować dla innych wyrazów.

      +
      +
      +
      +

      Liniowa zależność wektorów¶

      +
      +

      liniowa zależność wektorów

      +

      Wektory są liniowo niezależne gdy żadnego z nich nie można przedstawić w formie kombinacji liniowej pozostałych.

      +

      Przykładowo \(v = \alpha v_1 + \beta v_2 + \gamma v_3\) nie jest liniowo niezależny.

      +

      Biorąc pod uwagę inny przykład:

      +
      +\[\begin{split} +v = (5, 4, -18) \\ +v = 5 \hat{i} + 4 \hat{j} - 18 \hat{k} \\ +gdzie:\\ +\hat{i} = (1, 0, 0) \\ +\hat{j} = (0, 1, 0) \\ +\hat{k} = (0, 0, 1) \\ +\end{split}\]
      +
      +
      +

      Generatory

      +

      Zbiór generatorów to zbiór wektorów pozwalających na zapisanie każdego +innego wektora w danej podprzestrzeni.

      +
      +\[ +v = \alpha w_1 + \beta w_2 + ... + \theta w_n +\]
      +
        +
      • zbiór generatorów oznaczamy jako \(W\)

      • +
      • natomiast generowanÄ… przez nie podprzestrzeÅ„ \(limW\)

      • +
      +
      +
  • @@ -90,7 +167,17 @@

    Przestrzenie wektorowe