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@@ -52,9 +51,6 @@

Nawigacja

  • indeks
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  • Matematyka - dokumentacja »
  • @@ -492,7 +488,7 @@

    PÅ‚aszczyzny w przestrzeni \(\mat

    Przestrzenie wektorowe¶

    Przestrzeń Wektorowa

    -

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

    +

    (K, +, \(*\)) 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)

    • @@ -506,104 +502,6 @@

      Przestrzenie wektoroweWskazówka

      Elementy zbioru V nazywamy wektorami, a K - skalarami

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    -

    Informacja

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    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.

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    -
    -

    Podprzestrzenie liniowe¶

    -

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

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    -

    Informacja

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

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

    Ważne

    -

    0 musi należeć do podprzestrzeni

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

    -

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

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

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    Liniowa zależność wektorów¶

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    liniowa zależność wektorów

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

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

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    Biorąc pod uwagę inny przykład:

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

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    To tak jak w macierzach, gdzie jeden z rzędów można wyzerować za pomocą innych.

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    Informacja

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    Aby sprawdzić czy wektory są liniowo niezależne budujemy macierz -z wektorami wpisanymi w kolumny (tak aby poróœnywać x-owe współrzędne e.t.c.) i przyrównujemy do (0,0,…,0) -i obliczamy jej rząd.

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    Generatory

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

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

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

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    Twierdzenie o geneowaniu podprzestrzeni \(\mathbb{R}^n\)

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    Wektory \(v_1, v_2, ..., v_k\) generują przestrzeń \(\mathbb{R}^n\) (o n wymiarach) \(\Leftrightarrow\) \(rowA\) (rząd macierzy A) jest równy n. -Macierz A to taka macierz, która składa się w kolejno wpisanych w wierszach wektorach v.

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    Wskazówka

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    \(k \geq n\) tzn. że generatorów może być więcej niż wymiarów przestrzeni, jednak jeżeli tak jest -oznacza to że kilka z nich (\(k-n\)) jest liniowo zależnych.

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    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…:

    @@ -626,89 +524,30 @@

    Liniowa zależność wektorów -

    Baza¶

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    Układ wektorów (generatorów) można nazwać bazą, jeżeli:

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      -
    • sÄ… niezależne liniowo

    • -
    • generujÄ… przestrzeÅ„ wektorowÄ…

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    Baza

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    Podsumowując: układ n wektorów-generatorów liniowo niezależnych w przestrzeni n-wymiarowej jest bazą

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    Ważne

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    przestrzeń \(\left\{\bf{0}\right\}\) nie posiada bazy.

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    Wskazówka

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    Najprostszą bazą przestrzeni 3-wymiarowej jet układ wersorów \((\hat{j}, \hat{k}, \hat{l})\)

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    Reper Bazowy (aka Baza uporzÄ…dkowana)

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    To uporządkowany ciąg wektorów bazowych.

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    Wymiar przestrzeni wektorowej V

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    To liczba eleentów bazy i oznaczamy \(dimX\) gdzie X to przestrzeń.

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    Informacja

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    \(dim{0} = 0\)

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    Wskazówka

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    Wymiar podprzestrzeni jest mniejszy niż wymiar przestrzeni wyjściowej. -Jeżeli \(dimV = dimW \Rightarrow W = V\)

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    +

    Generatory

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    w przestrzei 2-wymiarowej generatorami są wersory \(\hat{i}\) i \(\hat{j}\), ponieważ każdy inny +wektor w tej przestrzeni można opisać w postaci kombinacji liniowej tych wektorów

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    Wskazówka

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    Współrzędnymi wektora nazywamy skalary \(v = \alpha_1 v_1 + \alpha_2 v_2 + ... + \alpha_n v_n\) -i zapisujemy jako \(v = [\alpha_1, \alpha_2, ..., \alpha_n]_B\)

    +
    +

    Układ niezależny

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    jeżeli \(a_1 v_1+ ... + a_n v_n = 0 \rightarrow a_1...a_n = 0\)

    +

    Oznacza to, że wszystkie elementy są istotne i żadnego nie można przedstawić jako kombinacji pozostałych.

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    Macierz Przejścia¶

    Informacja

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    Oznaczenie: \(P_{B \to B'}\)

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    Macierz przejścia

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    Wektor v należy do przestrzeni liniowej V.

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    W bazie B można zapisać go jako \(v = [x_1, x_2, ..., x_n]_B\) natomiast w bazie \(B'\): -\(v = [x_1', x_2', ..., x_n']_{B'}\)

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    W takim przypadku macierz przejścia można powiązać z wektorami w poszczególnych bazach jako: \(\bf{X = P_{B \to B'} X'}\).

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    X to wypisane w jdenej kolumnie współrzędne w bazie B a \(X'\) - w bazie \(B'\).

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    -\[\begin{split} -X = \begin{Bmatrix} -x_1 \\ -x_2 \\ -... \\ -x_n -\end{Bmatrix} -\quad -X' = \begin{Bmatrix} -x_1' \\ -x_2' \\ -... \\ -x_n' -\end{Bmatrix} -\end{split}\]
    -

    \(P_{B \to B'}\) to obraz azy \(B'\) w bazie \(B\)

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    -\[\begin{split} -P_{B \to B'} = \begin{Bmatrix} -b_{1_1} & b_{1_2} & ... & b_{1_n} \\ -b_{2_1} & b_{2_2} & ... & b_{2_n} \\ -... & ... & ... & ... \\ -b_{n_1} & b_{n_2} & ... & b_{n_n} \\ -\end{Bmatrix} -\end{split}\]
    +

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

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    Baza¶

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    Układ wektorów można nazwać bazą, jeżeli:

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      +
    • sÄ… niezależne liniowo

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    • generujÄ… przestrzeÅ„ wektorowÄ…

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    • ukÅ‚ad n wektorów liniowo niezależnych wprzestrzeni n-wymiarowej jest bazÄ…

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    Notatki z pliku notes/algebra/algebra_2023.11.21.2023.md

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