vault backup: 2026-03-05 16:54:10

This commit is contained in:
2026-03-05 16:54:10 +01:00
parent 4a5388ed66
commit c2f35479ac
12 changed files with 69896 additions and 743 deletions

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@@ -1,4 +1,5 @@
{ {
"cssTheme": "Catppuccin", "cssTheme": "Catppuccin",
"baseFontSize": 20 "baseFontSize": 20,
"nativeMenus": true
} }

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@@ -1,6 +1,7 @@
[ [
"obsidian-git", "obsidian-git",
"obsidian-style-settings", "obsidian-style-settings",
"obsidian-completr",
"obsidian-tikzjax", "obsidian-tikzjax",
"obsidian-completr" "obsidian-desmos"
] ]

View File

@@ -41,6 +41,7 @@ EdNL
Ez Ez
Exemple Exemple
Element Element
Endast
linjärt linjärt
ller ller
linjär linjär
@@ -75,6 +76,10 @@ lyfter
linjära linjära
lösningarna lösningarna
lönsing lönsing
längd
läst
leka
liksidiga
ekvationssystem ekvationssystem
en en
ekvationer ekvationer
@@ -118,6 +123,11 @@ endast
egenvektorer egenvektorer
exakt exakt
entydig entydig
entydligt
enhetsmatrisen
enher
enhet
enheter
med med
moam moam
matris matris
@@ -155,6 +165,7 @@ multiplicitet
mot mot
möjliga möjliga
matriserns matriserns
medger
reella reella
rella rella
rektagulär rektagulär
@@ -239,6 +250,10 @@ kolunrummet
kärna kärna
kärnrum kärnrum
kolomn kolomn
korninatsystemet
kombination
korndinaterna
kordinater
är är
än än
ändpunkten ändpunkten
@@ -302,6 +317,14 @@ sammanfaller
shcema shcema
schemat schemat
shcemat shcemat
ska
ste
skrivas
standerndbasen
standerdbasen
signerade
sägs
späns
av av
alla alla
allmänt allmänt
@@ -341,6 +364,8 @@ alltid
antaliet antaliet
antingen antingen
aldrig aldrig
area
arean
där där
det det
den den
@@ -420,6 +445,8 @@ index
ich ich
ibland ibland
ingen ingen
inversom
istället
variabler variabler
vatiabler vatiabler
vatiable vatiable
@@ -448,6 +475,14 @@ vanliga
vet vet
vata vata
vektorer vektorer
varja
vektoter
vekrje
vidare
vektorerna
varandra
vektoerna
viktigt
och och
om om
ordning ordning
@@ -476,6 +511,16 @@ ojämt
ohc ohc
oberoende oberoende
overrightarrow overrightarrow
ortagonal
olika
ortekonala
ortogonal
ortogonala
ortognala
ortiginal
oss
ordingen
orienterad
hat hat
herstamade herstamade
här här
@@ -513,6 +558,7 @@ gG
general general
genom genom
gra gra
gälla
för för
förekommer förekommer
första första
@@ -547,6 +593,7 @@ find
finnas finnas
fortsätning fortsätning
fira fira
fallet
term term
tal tal
till till
@@ -588,6 +635,10 @@ tänkas
tvp tvp
tredhe tredhe
ty ty
tt
tirangel
tar
triageln
ut ut
utgöt utgöt
under under
@@ -605,6 +656,10 @@ uppfyller
utvald utvald
upprepas upprepas
uppn uppn
up
utgörs
underförst
uo
HL HL
Hur Hur
HmE HmE
@@ -635,6 +690,7 @@ Similarly
Som Som
SATS SATS
Samma Samma
Standerdbasen
börjar börjar
bestämmer bestämmer
befiner befiner
@@ -661,6 +717,8 @@ beroende
byten byten
bort bort
bestämnda bestämnda
bas
beräknar
Ur Ur
Under Under
Uk Uk
@@ -691,6 +749,11 @@ parytor
polynom polynom
produkten produkten
prisis prisis
parallellogramet
pratar
parallellogramen
positiv
parallella
Alla Alla
Antigen Antigen
Avslutande Avslutande
@@ -704,6 +767,8 @@ At
Aa Aa
AT AT
Användiongs Användiongs
Anars
Areabyte
Oändligt Oändligt
Om Om
OBS OBS
@@ -788,6 +853,7 @@ Then
Transponering Transponering
Transponanten Transponanten
Transponaten Transponaten
Tv
Falsk Falsk
För För
Funktionen Funktionen
@@ -816,6 +882,8 @@ Koraste
KZ KZ
Koordinatrummet Koordinatrummet
Kallas Kallas
Kom
Kolumnerna
Primärfunktioner Primärfunktioner
Produkt Produkt
Paramaterformen Paramaterformen
@@ -847,11 +915,13 @@ It
In In
Inverse Inverse
Imdermatrosem Imdermatrosem
Ih
Bestäm Bestäm
Betäkning Betäkning
Bmm Bmm
BD BD
BEVIS BEVIS
Betrakta
öppet öppet
över över
cos cos
@@ -913,6 +983,7 @@ jmm
jS jS
jjj jjj
jämnt jämnt
jobbar
XmE XmE
XG XG
Xg Xg

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@@ -0,0 +1,8 @@
{
"version": "0.6.8",
"renderer": true,
"cache": {
"enabled": true,
"location": "Memory"
}
}

65724
.obsidian/plugins/obsidian-desmos/main.js vendored Normal file

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@@ -0,0 +1,8 @@
{
"id": "obsidian-desmos",
"name": "Desmos",
"version": "0.6.8",
"minAppVersion": "0.9.12",
"description": "Embed Desmos graphs into your notes",
"author": "Nigecat"
}

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File diff suppressed because one or more lines are too long

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@@ -6,5 +6,5 @@
"description": "Integrate Git version control with automatic backup and other advanced features.", "description": "Integrate Git version control with automatic backup and other advanced features.",
"isDesktopOnly": false, "isDesktopOnly": false,
"fundingUrl": "https://ko-fi.com/vinzent", "fundingUrl": "https://ko-fi.com/vinzent",
"version": "2.35.1" "version": "2.38.0"
} }

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@@ -8,6 +8,15 @@
} }
} }
.git-signs-gutter {
.cm-gutterElement {
/* Needed to align the sign properly for different line heigts. Such as
* when having a heading or list item.
*/
padding-top: 0 !important;
}
}
.workspace-leaf-content[data-type="git-view"] .button-border { .workspace-leaf-content[data-type="git-view"] .button-border {
border: 2px solid var(--interactive-accent); border: 2px solid var(--interactive-accent);
border-radius: var(--radius-s); border-radius: var(--radius-s);
@@ -72,6 +81,11 @@
height: 100%; height: 100%;
} }
/* Re-enable wrapping of nav buttns to prevent overflow on smaller screens #*/
.workspace-drawer .git-view .nav-buttons-container {
flex-wrap: wrap;
}
.git-tools { .git-tools {
display: flex; display: flex;
margin-left: auto; margin-left: auto;
@@ -129,444 +143,401 @@
color: var(--text-accent); color: var(--text-accent);
} }
.workspace-leaf-content[data-type="diff-view"] .d2h-d-none { /* ====== diff2html ======
display: none; The following styles are adapted from the obsidian-version-history plugin by
} @kometenstaub https://github.com/kometenstaub/obsidian-version-history-diff/blob/main/src/styles.scss
which itself is adapted from the diff2html library with the following original license:
.workspace-leaf-content[data-type="diff-view"] .d2h-wrapper {
text-align: left; https://github.com/rtfpessoa/diff2html/blob/master/LICENSE.md
}
Copyright 2014-2016 Rodrigo Fernandes https://rtfpessoa.github.io/
.workspace-leaf-content[data-type="diff-view"] .d2h-file-header {
background-color: var(--background-primary); Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated
border-bottom: 1px solid var(--interactive-accent); documentation files (the "Software"), to deal in the Software without restriction, including without limitation the
font-family: var(--font-monospace); rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit
height: 35px; persons to whom the Software is furnished to do so, subject to the following conditions:
padding: 5px 10px;
} The above copyright notice and this permission notice shall be included in all copies or substantial portions of the
Software.
.workspace-leaf-content[data-type="diff-view"] .d2h-file-header,
.workspace-leaf-content[data-type="diff-view"] .d2h-file-stats { THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE
display: -webkit-box; WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR
display: -ms-flexbox; COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR
display: flex; OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
} */
.workspace-leaf-content[data-type="diff-view"] .d2h-file-stats { .theme-dark,
font-size: 14px; .theme-light {
margin-left: auto; --git-delete-bg: #ff475040;
} --git-delete-hl: #96050a75;
--git-insert-bg: #68d36840;
.workspace-leaf-content[data-type="diff-view"] .d2h-lines-added { --git-insert-hl: #23c02350;
border: 1px solid #b4e2b4; --git-change-bg: #ffd55840;
border-radius: 5px 0 0 5px; --git-selected: #3572b0;
color: #399839;
padding: 2px; --git-delete: #c33;
text-align: right; --git-insert: #399839;
vertical-align: middle; --git-change: #d0b44c;
} --git-move: #3572b0;
}
.workspace-leaf-content[data-type="diff-view"] .d2h-lines-deleted {
border: 1px solid #e9aeae; .git-diff {
border-radius: 0 5px 5px 0; .d2h-d-none {
color: #c33; display: none;
margin-left: 1px; }
padding: 2px; .d2h-wrapper {
text-align: left; text-align: left;
vertical-align: middle; border-radius: 0.25em;
} overflow: auto;
}
.workspace-leaf-content[data-type="diff-view"] .d2h-file-name-wrapper { .d2h-file-header.d2h-file-header {
-webkit-box-align: center; background-color: var(--background-secondary);
-ms-flex-align: center; border-bottom: 1px solid var(--background-modifier-border);
align-items: center; font-family:
display: -webkit-box; Source Sans Pro,
display: -ms-flexbox; Helvetica Neue,
display: flex; Helvetica,
font-size: 15px; Arial,
width: 100%; sans-serif;
} height: 35px;
padding: 5px 10px;
.workspace-leaf-content[data-type="diff-view"] .d2h-file-name { }
overflow-x: hidden; .d2h-file-header,
text-overflow: ellipsis; .d2h-file-stats {
white-space: nowrap; display: -webkit-box;
} display: -ms-flexbox;
display: flex;
.workspace-leaf-content[data-type="diff-view"] .d2h-file-wrapper { }
border: 1px solid var(--background-modifier-border); .d2h-file-header {
border-radius: 3px; display: none;
margin-bottom: 1em; }
} .d2h-file-stats {
font-size: 14px;
.workspace-leaf-content[data-type="diff-view"] .d2h-file-collapse { margin-left: auto;
-webkit-box-pack: end; }
-ms-flex-pack: end; .d2h-lines-added {
-webkit-box-align: center; border: 1px solid var(--color-green);
-ms-flex-align: center; border-radius: 5px 0 0 5px;
align-items: center; color: var(--color-green);
border: 1px solid var(--background-modifier-border); padding: 2px;
border-radius: 3px; text-align: right;
cursor: pointer; vertical-align: middle;
display: none; }
font-size: 12px; .d2h-lines-deleted {
justify-content: flex-end; border: 1px solid var(--color-red);
padding: 4px 8px; border-radius: 0 5px 5px 0;
} color: var(--color-red);
margin-left: 1px;
.workspace-leaf-content[data-type="diff-view"] .d2h-file-collapse.d2h-selected { padding: 2px;
background-color: #c8e1ff; text-align: left;
} vertical-align: middle;
}
.workspace-leaf-content[data-type="diff-view"] .d2h-file-collapse-input { .d2h-file-name-wrapper {
margin: 0 4px 0 0; -webkit-box-align: center;
} -ms-flex-align: center;
align-items: center;
.workspace-leaf-content[data-type="diff-view"] .d2h-diff-table { display: -webkit-box;
border-collapse: collapse; display: -ms-flexbox;
font-family: Menlo, Consolas, monospace; display: flex;
font-size: 13px; font-size: 15px;
width: 100%; width: 100%;
} }
.d2h-file-name {
.workspace-leaf-content[data-type="diff-view"] .d2h-files-diff { overflow: hidden;
width: 100%; text-overflow: ellipsis;
} white-space: nowrap;
color: var(--text-normal);
.workspace-leaf-content[data-type="diff-view"] .d2h-file-diff { font-size: var(--h5-size);
overflow-y: hidden; }
} .d2h-file-wrapper {
border: 1px solid var(--background-secondary-alt);
.workspace-leaf-content[data-type="diff-view"] .d2h-file-side-diff { border-radius: 3px;
display: inline-block; margin-bottom: 1em;
margin-bottom: -8px; max-height: 100%;
margin-right: -4px; }
overflow-x: scroll; .d2h-file-collapse {
overflow-y: hidden; -webkit-box-pack: end;
width: 50%; -ms-flex-pack: end;
} -webkit-box-align: center;
-ms-flex-align: center;
.workspace-leaf-content[data-type="diff-view"] .d2h-code-line { align-items: center;
padding: 0 8em; border: 1px solid var(--background-secondary-alt);
} border-radius: 3px;
cursor: pointer;
.workspace-leaf-content[data-type="diff-view"] .d2h-code-line, display: none;
.workspace-leaf-content[data-type="diff-view"] .d2h-code-side-line { font-size: 12px;
display: inline-block; justify-content: flex-end;
-webkit-user-select: none; padding: 4px 8px;
-moz-user-select: none; }
-ms-user-select: none; .d2h-file-collapse.d2h-selected {
user-select: none; background-color: var(--git-selected);
white-space: nowrap; }
width: 100%; .d2h-file-collapse-input {
} margin: 0 4px 0 0;
}
.workspace-leaf-content[data-type="diff-view"] .d2h-code-side-line { .d2h-diff-table {
padding: 0 4.5em; border-collapse: collapse;
} font-family: var(--font-monospace);
font-size: var(--code-size);
.workspace-leaf-content[data-type="diff-view"] .d2h-code-line-ctn { width: 100%;
word-wrap: normal; }
background: none; .d2h-files-diff {
display: inline-block; width: 100%;
padding: 0; }
-webkit-user-select: text; .d2h-file-diff {
-moz-user-select: text; /*
-ms-user-select: text; overflow-y: scroll;
user-select: text; */
vertical-align: middle; border-radius: 5px;
white-space: pre; font-size: var(--font-text-size);
width: 100%; line-height: var(--line-height-normal);
} }
.d2h-file-side-diff {
.theme-light .workspace-leaf-content[data-type="diff-view"] .d2h-code-line del, display: inline-block;
.theme-light margin-bottom: -8px;
.workspace-leaf-content[data-type="diff-view"] margin-right: -4px;
.d2h-code-side-line overflow-x: scroll;
del { overflow-y: hidden;
background-color: #ffb6ba; width: 50%;
} }
.d2h-code-line {
.theme-dark .workspace-leaf-content[data-type="diff-view"] .d2h-code-line del, padding-left: 6em;
.theme-dark padding-right: 1.5em;
.workspace-leaf-content[data-type="diff-view"] }
.d2h-code-side-line .d2h-code-line,
del { .d2h-code-side-line {
background-color: #8d232881; display: inline-block;
} -webkit-user-select: none;
-moz-user-select: none;
.workspace-leaf-content[data-type="diff-view"] .d2h-code-line del, -ms-user-select: none;
.workspace-leaf-content[data-type="diff-view"] .d2h-code-line ins, user-select: none;
.workspace-leaf-content[data-type="diff-view"] .d2h-code-side-line del, white-space: nowrap;
.workspace-leaf-content[data-type="diff-view"] .d2h-code-side-line ins { width: 100%;
border-radius: 0.2em; }
display: inline-block; .d2h-code-side-line {
margin-top: -1px; /* needed to be changed */
text-decoration: none; padding-left: 0.5em;
vertical-align: middle; padding-right: 0.5em;
} }
.d2h-code-line-ctn {
.theme-light .workspace-leaf-content[data-type="diff-view"] .d2h-code-line ins, word-wrap: normal;
.theme-light background: none;
.workspace-leaf-content[data-type="diff-view"] display: inline-block;
.d2h-code-side-line padding: 0;
ins { -webkit-user-select: text;
background-color: #97f295; -moz-user-select: text;
text-align: left; -ms-user-select: text;
} user-select: text;
vertical-align: middle;
.theme-dark .workspace-leaf-content[data-type="diff-view"] .d2h-code-line ins, width: 100%;
.theme-dark /* only works for line-by-line */
.workspace-leaf-content[data-type="diff-view"] white-space: pre-wrap;
.d2h-code-side-line }
ins { .d2h-code-line del,
background-color: #1d921996; .d2h-code-side-line del {
text-align: left; background-color: var(--git-delete-hl);
} color: var(--text-normal);
}
.workspace-leaf-content[data-type="diff-view"] .d2h-code-line-prefix { .d2h-code-line del,
word-wrap: normal; .d2h-code-line ins,
background: none; .d2h-code-side-line del,
display: inline; .d2h-code-side-line ins {
padding: 0; border-radius: 0.2em;
white-space: pre; display: inline-block;
} margin-top: -1px;
text-decoration: none;
.workspace-leaf-content[data-type="diff-view"] .line-num1 { vertical-align: middle;
float: left; }
} .d2h-code-line ins,
.d2h-code-side-line ins {
.workspace-leaf-content[data-type="diff-view"] .line-num1, background-color: var(--git-insert-hl);
.workspace-leaf-content[data-type="diff-view"] .line-num2 { text-align: left;
-webkit-box-sizing: border-box; }
box-sizing: border-box; .d2h-code-line-prefix {
overflow: hidden; word-wrap: normal;
padding: 0 0.5em; background: none;
text-overflow: ellipsis; display: inline;
width: 3.5em; padding: 0;
} white-space: pre;
}
.workspace-leaf-content[data-type="diff-view"] .line-num2 { .line-num1 {
float: right; float: left;
} }
.line-num1,
.workspace-leaf-content[data-type="diff-view"] .d2h-code-linenumber { .line-num2 {
background-color: var(--background-primary); -webkit-box-sizing: border-box;
border: solid var(--background-modifier-border); box-sizing: border-box;
border-width: 0 1px; overflow: hidden;
-webkit-box-sizing: border-box; /*
box-sizing: border-box; padding: 0 0.5em;
color: var(--text-muted); */
cursor: pointer; text-overflow: ellipsis;
display: inline-block; width: 2.5em;
position: absolute; padding-left: 0;
text-align: right; }
width: 7.5em; .line-num2 {
} float: right;
}
.workspace-leaf-content[data-type="diff-view"] .d2h-code-linenumber:after { .d2h-code-linenumber {
content: "\200b"; background-color: var(--background-primary);
} border: solid var(--background-modifier-border);
border-width: 0 1px;
.workspace-leaf-content[data-type="diff-view"] .d2h-code-side-linenumber { -webkit-box-sizing: border-box;
background-color: var(--background-primary); box-sizing: border-box;
border: solid var(--background-modifier-border); color: var(--text-faint);
border-width: 0 1px; cursor: pointer;
-webkit-box-sizing: border-box; display: inline-block;
box-sizing: border-box; position: absolute;
color: var(--text-muted); text-align: right;
cursor: pointer; width: 5.5em;
display: inline-block; }
overflow: hidden; .d2h-code-linenumber:after {
padding: 0 0.5em; content: "\200b";
position: absolute; }
text-align: right; .d2h-code-side-linenumber {
text-overflow: ellipsis; background-color: var(--background-primary);
width: 4em; border: solid var(--background-modifier-border);
} border-width: 0 1px;
-webkit-box-sizing: border-box;
.workspace-leaf-content[data-type="diff-view"] .d2h-diff-tbody tr { box-sizing: border-box;
position: relative; color: var(--text-faint);
} cursor: pointer;
overflow: hidden;
.workspace-leaf-content[data-type="diff-view"] .d2h-code-side-linenumber:after { padding: 0 0.5em;
content: "\200b"; text-align: right;
} text-overflow: ellipsis;
width: 4em;
.workspace-leaf-content[data-type="diff-view"] .d2h-code-side-emptyplaceholder, /* needed to be changed */
.workspace-leaf-content[data-type="diff-view"] .d2h-emptyplaceholder { display: table-cell;
background-color: var(--background-primary); position: relative;
border-color: var(--background-modifier-border); }
} .d2h-code-side-linenumber:after {
content: "\200b";
.workspace-leaf-content[data-type="diff-view"] .d2h-code-line-prefix, }
.workspace-leaf-content[data-type="diff-view"] .d2h-code-linenumber, .d2h-code-side-emptyplaceholder,
.workspace-leaf-content[data-type="diff-view"] .d2h-code-side-linenumber, .d2h-emptyplaceholder {
.workspace-leaf-content[data-type="diff-view"] .d2h-emptyplaceholder { background-color: var(--background-primary);
-webkit-user-select: none; border-color: var(--background-modifier-border);
-moz-user-select: none; }
-ms-user-select: none; .d2h-code-line-prefix,
user-select: none; .d2h-code-linenumber,
} .d2h-code-side-linenumber,
.d2h-emptyplaceholder {
.workspace-leaf-content[data-type="diff-view"] .d2h-code-linenumber, -webkit-user-select: none;
.workspace-leaf-content[data-type="diff-view"] .d2h-code-side-linenumber { -moz-user-select: none;
direction: rtl; -ms-user-select: none;
} user-select: none;
}
.theme-light .workspace-leaf-content[data-type="diff-view"] .d2h-del { .d2h-code-linenumber,
background-color: #fee8e9; .d2h-code-side-linenumber {
border-color: #e9aeae; direction: rtl;
} }
.d2h-del {
.theme-light .workspace-leaf-content[data-type="diff-view"] .d2h-ins { background-color: var(--git-delete-bg);
background-color: #dfd; border-color: var(--git-delete-hl);
border-color: #b4e2b4; }
} .d2h-ins {
background-color: var(--git-insert-bg);
.theme-dark .workspace-leaf-content[data-type="diff-view"] .d2h-del { border-color: var(--git-insert-hl);
background-color: #521b1d83; }
border-color: #691d1d73; .d2h-info {
} background-color: var(--background-primary);
border-color: var(--background-modifier-border);
.theme-dark .workspace-leaf-content[data-type="diff-view"] .d2h-ins { color: var(--text-faint);
background-color: rgba(30, 71, 30, 0.5); }
border-color: #13501381; .d2h-del,
} .d2h-ins,
.d2h-file-diff .d2h-change {
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73
Area och Basbyte.md Normal file
View File

@@ -0,0 +1,73 @@
```desmos-graph
left=-5; right=5;
top=5; bottom=-5;
---
([0,0],[0,1])
([0,1],[0,0])
0 < y < 1 {0 < x < 1}
```
*En area enher av parallellogramet som spänns up av vektorerna. Standerdbasen $\overrightarrow{e_1},\;\overrightarrow{e_2}$ utgörs av korndinaterna av* $$\begin{bmatrix}
1&0\\0&1
\end{bmatrix}$$
**DEF**: *En "standerd area enhet" är lika med talet $\det{I}=1$. Om det är underförstått att vi jobbar med standerdbasen, då pratar vi endast om "area enheter".*
**DEF**: *Den signerade arean (dvs. arean med signerade + eller -) av parallellogramen som spänns uo av vektoerna* $$\overrightarrow{u}=(u_1,\;u_2),\;\overrightarrow{v}=(v_1,\;v_2)\in\mathbb{R}^2$$*är leka med determinanten av matrisen vars kolumner utgörs av $\overrightarrow{u}$ och $\overrightarrow{v}$*
*Om vi har en tirangel istället, få tar vi $\frac12$ av den här determinanten*
**OBS**: *ordingen av $\overrightarrow{u}$ och $\overrightarrow{v}$ är viktigt:*$$\underset{\substack{\parallel\\u_1v_2-v_1u_2}}{\det(\begin{bmatrix}
u_1&v_1\\u_2&v_2
\end{bmatrix}}=-1\underset{\substack{\parallel\\v_1u_2-u_1v_2}}{\det(\begin{bmatrix}
v_1&u_1\\v_2&u_2
\end{bmatrix}}$$
**DEF** *Två vektorer $\overrightarrow{u},\;\overrightarrow{v}$ sägs vara positiv orienterad om den signerade arean som späns upp av $\overrightarrow{u}$ och $\overrightarrow{v}$ är positiv*
**OBS** *Om $\overrightarrow{u}$ och $\overrightarrow{v}$ är parallella, då*$$\det(\underset{\substack{
\wedge\\\parallel\\\vee\\
\text{parallellogramen som spänns up av $\overrightarrow{u}$ och $\overrightarrow{v}$ har area }0
}}{\begin{bmatrix}
u_1&v_1\\
u_2&v_2
\end{bmatrix}})=0\Leftrightarrow\text{}\text{kolumnerna är linjärt levande}$$
```desmos-graph
left=-1; right=5;
top=1; bottom=-1;
---
(1,0.1)|blue|hidden|label:`\overrightarrow{v}`
(3,0.1)|green|hidden|label:`\overrightarrow{u}`
([0,2],[0,0])|blue
([0,4],[0,0])|green
```
[Graph of a triangle area]
*Area av den liksidiga triageln*$$\frac12\det(A)\frac12\times\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{4}$$
**Areabyte**:
- **Kordinater**: $$I\times\begin{bmatrix}
\zeta_1\\\zeta_2\\\zeta_3
\end{bmatrix}=A\times\begin{bmatrix}
\alpha_1\\\alpha_2\\\alpha_3
\end{bmatrix}$$
- **Area**: *Om vi hade en area av $X$ a.e. innan basbyte, då har vi $\det{A}\times{X}$ a.e. efter basbyte.*
- **Volym**: *x v.e. före basbyte $\Rightarrow$ $\det(A)\times{X}$ a.e. efter basbyte.*
**OBS**:
- *Area av triangle $=\frac12$ area av parallellogram*
- *Volum av tetraheder $=\frac13$ volum av parallellepiod*
- *4-d volum av 4-d tetrahden $=\frac1{24}$ 4-d volum av 4-d parallelopipod*
**SATS**: *Låt $A$ vara en $m\times{n}$ ortogonal matris. Då är $\operatorname{def}(A)$ antigen $+1$ eller $-1$.*
**BEVIS**:
- *För ortogonala matriser är $A^{-1}=A^T$*
- *$\det(A)=\det(A^T)$*
- *$\operatorname{def}(AB)=\operatorname{def}(A)\times\operatorname{def}(B)$*
$\Rightarrow{A}\times{A^T}=I\Rightarrow\det(AA^T)=\det(I)\Rightarrow\det(A)\times\det{A^T}=1\Rightarrow\operatorname{def}(A)^2=1\Rightarrow\operatorname{def}(A)\text{är }+1\text{ eller }-1$
**OBS**: *Om vi har en $m\times{n}$ matris $A$, då är $\det(A)$ lika med den $m-$dimensonella volymen av figuren som spenns up av matrises kolumner*
**EX**: $$\begin{bmatrix}
1&0\\0&1
\end{bmatrix}\rightarrow\begin{bmatrix}
1&\frac12\\
0&\frac{\sqrt{3}}2
\end{bmatrix}\Rightarrow\text{svårt att beskriva}$$
[ ]
**FAKTA**: *Om $A$ är en ortogonal matris, då är skälärprodukten nellan två vektorer samma i så val den gamla basen som den nya basen*