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FHTW.typ
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FHTW.typ
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#import "@preview/cetz:0.4.2"
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#import "LineareAlgebraGrafiken.typ": *
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#set page(
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margin: 2cm,
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paper: "a4"
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)
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#title("Mathematik FHTW Berlin")
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= Lineare Algebra
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== Vektoren
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=== Veranschaulichung von Vektoren in der Ebene
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#show math.equation: set text(font: "New Computer Modern Sans Math", size: 12pt)
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#show math.equation.where(block: false): it => math.display(it)
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//#set page(width: auto, height: auto, margin: .5cm)
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#table(
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columns: 3,
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stroke: none,
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[#abb1()],[#h(1cm)], [#abb2()]
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)//&#show math.equation: block.with(fill: white, inset: 1pt)
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$arrow(a)=(a_1, a_2)$, auch üblich $arrow(a)=vec(a_1, a_2) arrow.l$ geordnetes Paar
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=== Menge aller Vektoren in der Ebene
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Die Menge aller Vektoren in der Ebene heißt $RR^2$; dabei ist $RR$ die Menge der reellen Zahlen.
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Also: $RR={arrow(a)=(a_1,a_2) | a_1 in RR, a_2 in RR }$
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=== Addition von Vektoren in der Ebene
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- liefert wieder einen Vektor
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- rechnerisch:
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- $arrow(a)=(a_1, a_2)$, $arrow(b)=(b_1, b_2)$
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- $arrow(a)+arrow(b)=(a_1+b_1, a_2+b_2)$
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#abb3()
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/*1.1.2 Menge aller Vektoren in der Ebene
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Die Menge aller Vektoren in der Ebene heißt $\mathbb{R}^2$; dabei ist $\mathbb{R}$ die Menge der reellen Zahlen. Also: $\mathbb{R}=\left\{\vec{a}=\left(a_1, a_2\right) \mid a_1 \in \mathbb{R}, a_2 \in \mathbb{R}\right\}$
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1.1.3 Addition von Vektoren in der Ebene
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Die Addition von Vektoren liefert als Ergebnis wieder einen Vektor.
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rechnerisch:
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$$
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\begin{aligned}
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& \vec{a}=\left(a_1, a_2\right), \vec{b}=\left(b_1, b_2\right) \\
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& \vec{a}+\vec{b}=\left(\left(a_1+b_1, a_2+b_2\right)\right)
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\end{aligned}
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$$
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zeichnerisch:
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Rechenregeln:
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$$
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\begin{aligned}
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& \vec{a}+\vec{b}=\vec{b}+\vec{a} \\
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& \vec{a}+(\vec{b}+\vec{c})=(\vec{a}+\vec{b})+\vec{c}
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\end{aligned}
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$$*/
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152
LineareAlgebraGrafiken.typ
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152
LineareAlgebraGrafiken.typ
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#import "@preview/cetz:0.3.3" //Grafiken
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#let abb1() = {
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cetz.canvas(length: 3cm, {
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import cetz.draw: *
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set-style(
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mark: (fill: black, scale: 2),
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stroke: (thickness: 0.4pt, cap: "round"),
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angle: (
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radius: 0.3,
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label-radius: .22,
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fill: green.lighten(80%),
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stroke: (paint: green.darken(50%))
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),
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content: (padding: 1pt)
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)
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grid((-.25, -.25), (2, 2 ), step: 0.5, stroke: gray + 0.2pt)
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line((-.25, 0), (1.85, 0), mark: (end: "stealth"))
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content((), $ x $, anchor: "west")
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line((0, -.25), (0, 1.85), mark: (end: "stealth"))
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content((), $ y $, anchor: "south")
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for (x, ct) in ((0, $ 0 $), (0.5, $ 1 $), (1, $ 2 $), (1.5, $ 3 $), (1.25, $a_1$)) {
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line((x, 3pt), (x, -3pt))
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content((), anchor: "north", ct)
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}
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for (y, ct) in ((0.5, $ 1 $), (1, $ 2 $), (1.5, $ 3 $), (1.25, $a_2$)) {
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line((3pt, y), (-3pt, y))
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content((), anchor: "east", ct)
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}
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//line((0,0), (1.25, 1.25),name: "vec",mark: (end: "stealth", fill: green.darken(30%)),
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//stroke: (paint: green.darken(30%), thickness: 1.25pt))
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line((0,0), (1.25, 1.25),name: "vec",stroke: (paint: red.darken(30%), thickness: 1.25pt))
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line((1.25,0), (1.25, 1.25), stroke: (dash: "dashed", paint: blue.darken(30%), thickness: 0.75pt))
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line((0,1.25), (1.25, 1.25), stroke: (dash: "dashed", paint: blue.darken(30%), thickness: 0.75pt))
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circle((1.25,1.25), radius: 0.025, fill: green.darken(30%), stroke: (paint: green.darken(30%), thickness: 1.25pt), name: "punkt")
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content("punkt.end", anchor: "south-west", padding: (left: 3mm, bottom: 2mm), text(green.darken(30%))[Punkt $(a_1, a_2)$])})
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}
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#let abb2() = {
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cetz.canvas(length: 3cm, {
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import cetz.draw: *
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set-style(
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mark: (fill: black, scale: 2),
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stroke: (thickness: 0.4pt, cap: "round"),
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angle: (
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radius: 0.3,
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label-radius: .22,
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fill: green.lighten(80%),
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stroke: (paint: green.darken(50%))
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),
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content: (padding: 1pt)
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)
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grid((-.25, -.25), (2, 2 ), step: 0.5, stroke: gray + 0.2pt)
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line((-.25, 0), (1.85, 0), mark: (end: "stealth"))
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content((), $ x $, anchor: "west")
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line((0, -.25), (0, 1.85), mark: (end: "stealth"))
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content((), $ y $, anchor: "south")
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for (x, ct) in ((0, $ 0 $), (0.5, $ 1 $), (1, $ 2 $), (1.5, $ 3 $), (1.25, $a_1$)) {
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line((x, 3pt), (x, -3pt))
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content((), anchor: "north", ct)
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}
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for (y, ct) in ((0.5, $ 1 $), (1, $ 2 $), (1.5, $ 3 $), (1.25, $a_2$)) {
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line((3pt, y), (-3pt, y))
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content((), anchor: "east", ct)
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}
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line((0,0), (1.25, 1.25),name: "vec",stroke: (paint: red.darken(30%), thickness: 1.25pt))
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line((1.25,0), (1.25, 1.25), stroke: (dash: "dashed", paint: blue.darken(30%), thickness: 0.75pt))
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line((0,1.25), (1.25, 1.25), stroke: (dash: "dashed", paint: blue.darken(30%), thickness: 0.75pt))
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line((0,0), (1.25, 1.25),name: "vec",mark: (end: "stealth", fill: green.darken(30%)), stroke: (paint: green.darken(30%), thickness: 1.25pt))
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content("vec.end", anchor: "north-west", padding: (left: 3mm, top: -2mm), text(green.darken(30%))[Vektor $arrow(a)$])
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})
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}
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#let abb3() = {
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cetz.canvas(length: 37mm,
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{
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import cetz.draw: *
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set-style(
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mark: (fill: black, scale: 2),
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stroke: (thickness: 0.4pt, cap: "round"),
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angle: (
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radius: 0.3,
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label-radius: .22,
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fill: green.lighten(80%),
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stroke: (paint: green.darken(50%))
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),
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content: (padding: 1pt)
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)
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grid((-.25, -.25), (2.5 , 2 ), step: 0.5, stroke: gray + 0.2pt)
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line((-.25, 0), (2.25, 0), mark: (end: "stealth"))
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content((), $ x $, anchor: "west")
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line((0, -.25), (0, 1.85), mark: (end: "stealth"))
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content((), $ y $, anchor: "south")
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for (x, ct) in ((0.25, $ 1 $), (0.5, $ 2$), (0.75, $ a_1 $),(1, $b_1$), (1.25,$5$), (1.5, $6$),(1.75, $a_1+b_1$)) {
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line((x, 3pt), (x, -3pt))
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content((), anchor: "north", ct)
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}
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for (y, ct) in ((0.25, $ 1 $), (0.5, $ a_2 $), (.75, $ 3 $), (1., $b_2$),(1.25,$5$), (1.5, $6$)) {
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line((3pt, y), (-3pt, y))
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content((), anchor: "east", ct)
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}
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line((0,0), (0.75, .5),name: "veca",mark: (end: "stealth", fill: blue.darken(30%)), stroke: (paint: blue.darken(30%), thickness: 0.75pt))
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content("veca.100%", anchor: "west", padding: (left:1mm, bottom:-5mm),text(blue.darken(30%))[$arrow(a)$])
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line((0,0), (1, 1),name: "vecb",mark: (end: "stealth", fill: red.darken(30%)), stroke: (paint: red.darken(30%), thickness: 0.75pt))
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content("vecb.80%", anchor: "east", padding: (left:0mm, bottom:5mm),text(red.darken(30%))[$arrow(b)$])
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//
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//line((0,0), (1.5, 1),name: "vecb",mark: (end: "stealth", fill: red), stroke: (paint: red, thickness: 0.75pt))
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})}
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