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Ekvations System.md
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- *Exakt-bestämnd*: *Lika många ekvationer som variabler* - *Exakt-bestämnd*: *Lika många ekvationer som variabler*
- *Över-bestämnd*: *Mera ekvationser än variabler* - *Över-bestämnd*: *Mera ekvationser än variabler*
- *Under-bestämnd*: *Mindre ekvationser än variablar* - *Under-bestämnd*: *Mindre ekvationser än variablar*
- **Ex**: **Exakt bestämd system/Entydlig lösning** - **Ex**:
1. $$\begin{aligned}\begin{aligned}x-2y+z&=&3\\2x+y&=&1\\3y+2z&=&2\end{aligned}\Rightarrow\begin{pmatrix}1&-2&1&|&3\\2&-1&0&|&1\\0&3&2&|&2\end{pmatrix}\\\begin{aligned}\frac12R_2\rightarrow{R_2}\\\xrightarrow{}\end{aligned}\begin{pmatrix}1&-2&1&|&3\\1&-\frac12&0&|&\frac12\\0&3&2&|&2\end{pmatrix}\begin{aligned}R_2-R_1\rightarrow{R_2}\\\xrightarrow{}\end{aligned}\begin{pmatrix}1&-2&1&|&3\\0&\frac32&-1&|&-\frac52\\0&3&2&|&2\end{pmatrix}\\\begin{aligned}\frac32R_2\rightarrow{R_2}\\\xrightarrow{}\end{aligned}\begin{pmatrix}1&-2&1&|&3\\0&1&-\frac23&|&-\frac53\\0&3&2&|&2\end{pmatrix}\begin{aligned}R_3-3R_2\rightarrow{R_3}\\\xrightarrow{}\end{aligned}\begin{pmatrix}1&-2&1&|&3\\0&1&-\frac23&|&-\frac53\\0&0&4&|&7\end{pmatrix}\\\begin{aligned}\frac14R_3\rightarrow{R_3}\\\xrightarrow{}\end{aligned}\begin{pmatrix}1&-2&1&|&3\\0&1&-\frac23&|&-\frac53\\0&0&1&|&\frac74\end{pmatrix}\end{aligned}$$ *Vi hat nu bekräftat att ekvations systemet har en entydlig lösning*$$\begin{aligned}1\times{z}=\frac74\Longrightarrow{z=\frac74}\\1\times{y}-\frac23z=-\frac53\Rightarrow{y}=\frac23z-\frac53=\end{aligned}$$ 1. **Exakt bestämd system/Entydlig lösning**$$\begin{aligned}\begin{aligned}x-2y+z&=&3\\2x+y&=&1\\3y+2z&=&2\end{aligned}\Rightarrow\begin{pmatrix}1&-2&1&|&3\\2&-1&0&|&1\\0&3&2&|&2\end{pmatrix}\\\begin{aligned}\frac12R_2\rightarrow{R_2}\\\xrightarrow{}\end{aligned}\begin{pmatrix}1&-2&1&|&3\\1&-\frac12&0&|&\frac12\\0&3&2&|&2\end{pmatrix}\begin{aligned}R_2-R_1\rightarrow{R_2}\\\xrightarrow{}\end{aligned}\begin{pmatrix}1&-2&1&|&3\\0&\frac32&-1&|&-\frac52\\0&3&2&|&2\end{pmatrix}\\\begin{aligned}\frac32R_2\rightarrow{R_2}\\\xrightarrow{}\end{aligned}\begin{pmatrix}1&-2&1&|&3\\0&1&-\frac23&|&-\frac53\\0&3&2&|&2\end{pmatrix}\begin{aligned}R_3-3R_2\rightarrow{R_3}\\\xrightarrow{}\end{aligned}\begin{pmatrix}1&-2&1&|&3\\0&1&-\frac23&|&-\frac53\\0&0&4&|&7\end{pmatrix}\\\begin{aligned}\frac14R_3\rightarrow{R_3}\\\xrightarrow{}\end{aligned}\begin{pmatrix}1&-2&1&|&3\\0&1&-\frac23&|&-\frac53\\0&0&1&|&\frac74\end{pmatrix}\end{aligned}$$
- *Vi hat nu bekräftat att ekvations systemet har en entydlig lösning*$$\begin{aligned}1\times{z}=\frac74\Longrightarrow{z=\frac74}\\1\times{y}-\frac23z=-\frac53\Rightarrow{y}=\frac23z-\frac53=\end{aligned}$$
2. **Exakt-bestämnd system/oändliga lösningar**$$\begin{aligned}\begin{aligned}x-2y+z&=&3\\2x-2y&=&1\\3x-4y+z&=&4\end{aligned}\Rightarrow\begin{pmatrix}1&-2&1&|&3\\2&-2&0&|&1\\3&-4&1&|&4\end{pmatrix}\begin{aligned}R_2-2R_1\rightarrow{R_2}\\R_3-3R_1\rightarrow{R_3}\\\xrightarrow{}\end{aligned}\begin{pmatrix}1&-2&1&|&3\\0&2&-2&|&-5\\0&2&-2&|&-5\end{pmatrix}\\\begin{aligned}R_3-R_2\rightarrow{R_3}\\\xrightarrow{}\end{aligned}\begin{pmatrix}1&-2&1&|&3\\0&2&-2&|&-5\\0&0&0&|&0\end{pmatrix}\begin{aligned}\frac12R_2\rightarrow{R_2}\\\xrightarrow{}\end{aligned}\begin{pmatrix}1&-2&1&|&3\\0&1&-1&|&-\frac52\\0&0&0&|&0\end{pmatrix}\Rightarrow\\\begin{aligned}\text{Vi har två pivåvariabler $x$ och $y$, och en fri variabel, $z$. Vi har altså oändligt många lösningar.}\end{aligned}\end{aligned}$$
- *Eftersom $z$ är en fri variabler kan $z=t$, och $t\in\mathbb{R}$. sampt* $$\begin{aligned}y-z=-\frac52\Rightarrow{y}=z-\frac52=t-\frac52\\x-2y+z=3\Rightarrow{x}=2y-z+3=2\left(t-\frac52\right)-t+3=t-2\\\end{aligned}$$
3. **Exakt-bestämnd system/Saknar lösningar** $$\begin{aligned}\begin{aligned}x-3y+2z&=&3\\x-2y&=&2\\2x-5y+2z&=&4\end{aligned}\Rightarrow\begin{pmatrix}1&-3&2&|&3\\1&-2&2&|&2\\2&-5&2&|&-4\end{pmatrix}\begin{aligned}R_2-R_1\rightarrow{R_2}\\R_3-2R_1\rightarrow{R_3}\\\xrightarrow{}\end{aligned}\begin{pmatrix}1&-3&2&|&3\\0&1&0&|&-1\\0&1&-2&|&-2\end{pmatrix}\\\begin{aligned}R_3-R_2\rightarrow{R_3}\\\xrightarrow{}\end{aligned}\pmatrix{1&-3&2&|&3\\0&1&2&|&-1\\0&0&0&|&-1}\end{aligned}$$
- **OBS** *Rad $2$ och $3$ säger att det skall vara $-2$ medans de int har samma $VL$, detta går inte! samt säger det $0x+0y+0z=-1\Leftrightarrow{0=-1}$*
- **Ex**: $$\begin{aligned}\begin{aligned}x_1-2x_2-3x_x&=&0\\x_1-x_4&=&-2\end{aligned}\\\\\Rightarrow\begin{pmatrix}1&-2&-3&0&|&0\\1&0&0&-1&|&-2\end{pmatrix}\end{aligned}$$ - **Ex**: $$\begin{aligned}\begin{aligned}x_1-2x_2-3x_x&=&0\\x_1-x_4&=&-2\end{aligned}\\\\\Rightarrow\begin{pmatrix}1&-2&-3&0&|&0\\1&0&0&-1&|&-2\end{pmatrix}\end{aligned}$$
- **Ex**: $$\left.\begin{aligned}x+2y-u+3v&=&2\\2x+3y+2z-2u+10v&=&0\\x+3y-2z-4u+2v&=&3\\\underbrace{-x-3y+2z+3u-v}_{\substack{\text{VL $4\times5$}\\\text{=20 platser i schemat}}}&=&\underbrace{-4}_{\substack{\text{HL $4$}\\\text{ platser}}}\\\end{aligned}\right.\Rightarrow\left(a\mid\overrightarrow{b}\right)=\begin{pmatrix}1&2&0&-1&3&|&2\\2&3&2&-2&10&|&0\\1&3&-2&-3&2&|&3\\-1&-3&2&3&1&|&-4\end{pmatrix}$$ - **Ex**: $$\left.\begin{aligned}x+2y-u+3v&=&2\\2x+3y+2z-2u+10v&=&0\\x+3y-2z-4u+2v&=&3\\\underbrace{-x-3y+2z+3u-v}_{\substack{\text{VL $4\times5$}\\\text{=20 platser i schemat}}}&=&\underbrace{-4}_{\substack{\text{HL $4$}\\\text{ platser}}}\\\end{aligned}\right.\Rightarrow\left(a\mid\overrightarrow{b}\right)=\begin{pmatrix}1&2&0&-1&3&|&2\\2&3&2&-2&10&|&0\\1&3&-2&-3&2&|&3\\-1&-3&2&3&1&|&-4\end{pmatrix}$$
*Hur räknar man med ett gauss schema? Man räknar med hjälp av elemäntera radoperationer:* *Hur räknar man med ett gauss schema? Man räknar med hjälp av elemäntera radoperationer:*