#LyX 2.3 created this file. 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\html_be_strict false \end_header \begin_body \begin_layout Title List s formulami za 1. kolokvij Linearne algebre \end_layout \begin_layout Author \noun on Anton Luka Šijanec \end_layout \begin_layout Date \begin_inset ERT status open \begin_layout Plain Layout \backslash today \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash newcommand \backslash euler{e} \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash begin{multicols}{2} \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Formula $\vec{u}\cdot\vec{u}=\vert\vert\vec{u}\vert\vert^{2}$ \end_inset , \begin_inset Formula $\left(\alpha\vec{u}+\beta\vec{v}\right)\cdot\vec{w}=\alpha\left(\vec{u}\cdot\vec{w}\right)+\beta\left(\vec{v},\vec{w}\right)$ \end_inset \end_layout \begin_layout Standard Paralelogramska ident.: \begin_inset Formula $\vert\vert\vec{u}+\vec{v}\vert\vert^{2}+\vert\vert\vec{u}-\vec{v}\vert\vert^{2}=2\vert\vert\vec{u}\vert\vert^{2}+2\vert\vert\vec{v}\vert\vert^{2}$ \end_inset \end_layout \begin_layout Standard Ploščina paralelograma: \begin_inset Formula $\vert\vert\vec{u}\times\vec{v}\vert\vert=\vert\vert\vec{u}\vert\vert\vert\vert\vec{v}\vert\vert\sin\phi$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula $<\vec{u},\vec{v}>=\vert\vert\vec{u}\vert\vert\vert\vert\vec{v}\vert\vert\cos\phi$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula $<\vec{u}\times\vec{v},\vec{u}>=0$ \end_inset , \begin_inset Formula $\vert\vert\vec{u}\times\vec{v}\vert\vert^{2}+<\vec{u},\vec{v}>^{2}=\vert\vert\vec{u}\vert\vert^{2}\vert\vert\vec{v}\vert\vert^{2}$ \end_inset \end_layout \begin_layout Standard Vol. ppd.: \begin_inset Formula $[u,v,w]==\vert\vert u\times v\vert\vert\cdot\vert\vert w\vert\vert\cos\phi$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula $\vec{u}\times\vec{u}=0$ \end_inset , \begin_inset Formula $\vec{u}\times\vec{v}=-\left(\vec{v}\times\vec{u}\right)$ \end_inset \end_layout \begin_layout Standard Linearnost \begin_inset Formula $\times$ \end_inset : \begin_inset Formula $\left(\alpha\vec{u}+\beta\vec{v}\right)\times\vec{w}=\alpha\left(\vec{u}\times\vec{w}\right)+\beta\left(\vec{v}\times\vec{w}\right)$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula $[u,v,w]=-[u,w,v]$ \end_inset , \begin_inset Formula $[u,v,w]=[w,u,v]=[v,w,u]$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula $\vec{r}=\vec{r_{0}}+t\vec{p},t\in\mathbb{R}\Longleftrightarrow x=x_{0}+tp_{1},y=y_{0}+tp_{2},z=z_{0}+tp_{3}$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula $\Longleftrightarrow t=\frac{x-x_{0}}{p_{1}}=\frac{y-y_{0}}{p_{2}}=\frac{z-z_{0}}{p_{3}}$ \end_inset :Normalna enačba \begin_inset Formula $\mathbb{R}^{3}$ \end_inset premice \end_layout \begin_layout Standard Projekcija \begin_inset Formula $\vec{r_{1}}$ \end_inset na \begin_inset Formula $\vec{r_{0}}+t\vec{p}$ \end_inset \begin_inset Formula $\coloneqq$ \end_inset \begin_inset Formula $\vec{r_{1}'}=\vec{r_{0}}+t'\vec{p}$ \end_inset in \begin_inset Formula $<\vec{r_{1}'}-\vec{r_{1}},\vec{p}>=0$ \end_inset \end_layout \begin_layout Standard Dvojni \begin_inset Formula $\times$ \end_inset : \begin_inset Formula $\vec{a}\times\left(\vec{b}\times\vec{c}\right)=\vec{b}\left(\vec{a}\cdot\vec{c}\right)-\vec{c}\left(\vec{a}\cdot\vec{b}\right)$ \end_inset \end_layout \begin_layout Standard Dvojni \begin_inset Formula $\times$ \end_inset : \begin_inset Formula $\left(\vec{a}\times\vec{b}\right)\times\vec{c}=\vec{b}\left(\vec{a}\cdot\vec{c}\right)-\vec{a}\left(\vec{b}\cdot\vec{c}\right)$ \end_inset \end_layout \begin_layout Standard Norm. e. ravnine: \begin_inset Formula $n_{1}x+n_{2}y+n_{3}z=d=n_{1}x_{0}+n_{2}y_{0}+n_{3}z_{0}$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula $\Longleftrightarrow<\vec{r}-\vec{r_{0}},\vec{n}>=0$ \end_inset , saj je \begin_inset Formula $\forall\vec{r}\in$ \end_inset ravnine \begin_inset Formula $:\left(\vec{r}-\vec{r_{0}}\right)\bot\vec{n}$ \end_inset \end_layout \begin_layout Standard Parametrična e. ravnine: \begin_inset Formula $\vec{r}=\vec{r_{0}}+s\vec{p}+t\vec{q},s\in\mathbb{R},q\in\mathbb{R}$ \end_inset \end_layout \begin_layout Standard Proj. \begin_inset Formula $\vec{r_{1}}=\vec{r_{0}}+s\vec{p}+t\vec{q}$ \end_inset velja \begin_inset Formula $<\vec{r_{1}'}-\vec{r_{1},\vec{p}}>=0=<\vec{r_{1}'}-\vec{r_{1},\vec{q}}>$ \end_inset \end_layout \begin_layout Standard sistem: \begin_inset Formula $s\vec{p}\cdot\vec{p}+t\vec{q}\cdot\vec{p}=\left(\vec{r_{1}}-\vec{r_{0}}\right)\cdot\vec{p}$ \end_inset ; \begin_inset Formula $s\vec{p}\cdot\vec{q}+t\vec{q}\cdot\vec{q}=\left(\vec{r_{1}}-\vec{r_{0}}\right)\cdot\vec{q}$ \end_inset \end_layout \begin_layout Standard Hiperravnino v \begin_inset Formula $\mathbb{R}^{n}$ \end_inset določa \begin_inset Formula $n$ \end_inset linearno neodvisnih vektorjev. \end_layout \begin_layout Standard \begin_inset Note Note status open \begin_layout Plain Layout Posplošena rešitev: \begin_inset Formula $\min\sum_{k=1}^{m}\left(a_{k,1}x_{1}+\ldots+a_{1,n}x_{n}-b_{k}\right)^{2}$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\Longleftrightarrow\min\vert\vert x_{1}\vec{a_{1}}+\ldots+x_{n}\vec{a_{n}}-\vec{b}\vert\vert^{2}$ \end_inset (proj \begin_inset Formula $\vec{b}$ \end_inset na hiperravnino) \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Formula $\left(AB\right)^{T}=B^{T}+A^{T}$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula $E_{ij}\left(\alpha\right)\coloneqq\texttt{i+=\ensuremath{\alpha}j}$ \end_inset , \begin_inset Formula $P_{ij}\coloneqq\texttt{i,j=j,i}$ \end_inset , \begin_inset Formula $E_{i}\left(\alpha\right)\coloneqq\texttt{i*=\ensuremath{\alpha}}$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula $E_{ij}\left(\alpha\right)^{-1}=E_{ij}\left(\alpha\right)$ \end_inset , \begin_inset Formula $P_{ij}^{-1}=P_{ji}$ \end_inset , \begin_inset Formula $E_{i}\left(\beta\right)^{-1}=E_{i}\left(\beta^{-1}\right)$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula $\nexists A_{m,n}^{-1}\Leftrightarrow A=0\Leftrightarrow m\not=n\Leftrightarrow\det A=0\Leftrightarrow A$ \end_inset ima \begin_inset Formula $\vec{0}$ \end_inset vrstico/stolpec \end_layout \begin_layout Paragraph Karakterizacija obrnljivih matrik \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash begin{multicols}{2} \end_layout \end_inset \end_layout \begin_layout Itemize \begin_inset Argument 1 status open \begin_layout Plain Layout label= \begin_inset Formula $\Leftrightarrow$ \end_inset \end_layout \end_inset \begin_inset Formula $\exists A^{-1}$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $\exists B\ni:BA=I$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $\exists B\ni:AB=I$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $\left(AX=0\Longrightarrow X=0\right)$ \end_inset \end_layout \begin_layout Itemize stolpci so ogrodje \end_layout \begin_layout Itemize \begin_inset Formula $\text{RKSO}\left(A\right)=I$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $\forall\vec{b}\exists\vec{x}\ni:A\vec{x}=\vec{b}$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $A=$ \end_inset produkt E. M. \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash end{multicols} \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Note Note status open \begin_layout Plain Layout \begin_inset Formula $\exists A^{-1}\Longleftrightarrow\exists B\ni:BA=I\Longleftrightarrow\exists B\ni:AB=I\Longleftrightarrow$ \end_inset stolpci so LN \begin_inset Formula $\Longleftrightarrow\left(AX=0\Longrightarrow X=0\right)\Longleftrightarrow$ \end_inset stolpci so ogrodje \begin_inset Formula $\Longleftrightarrow\text{RKSO}\left(A\right)=$ \end_inset \begin_inset Formula $I\Longleftrightarrow\forall\vec{b}\exists\vec{x}\ni:A\vec{x}=\vec{b}\Longleftrightarrow A=$ \end_inset produkt E.M. \end_layout \end_inset \end_layout \begin_layout Standard Matrični zapis sistema: \begin_inset Formula $A\vec{x}=\vec{b}$ \end_inset \end_layout \begin_layout Standard Najkrajša rešitev sistema \begin_inset Formula $\vec{x_{0}}\Leftarrow\vert\vert A\vec{x_{0}}-\vec{b}\vert\vert=\min\vert\vert A\vec{x}-\vec{b}\vert\vert$ \end_inset \end_layout \begin_layout Standard ... je običajna rešitev \begin_inset Formula $A^{T}A\vec{x}=A^{T}\vec{b}$ \end_inset \end_layout \begin_layout Standard Desno množenje z E. M. je manipulacija stoplcev. \end_layout \begin_layout Standard \begin_inset Formula $M/A\coloneqq D-CA^{-1}B$ \end_inset , \begin_inset Formula $M/D\coloneqq A-BD^{-1}C$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula \[ M^{-1}=\left[\begin{array}{cc} A & B\\ C & D \end{array}\right]^{-1}= \] \end_inset \end_layout \begin_layout Standard \begin_inset Formula \[ =\left[\begin{array}{cc} A^{-1}+A^{-1}B\left(M/A\right)^{-1}CA^{-1} & -A^{-1}B\left(M/A\right)^{-1}\\ -\left(M/A\right)^{-1}CA^{-1} & \left(M/A\right)^{-1} \end{array}\right]= \] \end_inset \begin_inset Formula \[ =\left[\begin{array}{cc} \left(M/D\right)^{-1} & -\left(M/D\right)^{-1}BD^{-1}\\ -D^{-1}C\left(M/D\right)^{-1} & D^{-1}+D^{-1}C\left(M/D\right)^{-1}BD^{-1} \end{array}\right] \] \end_inset \end_layout \begin_layout Standard \begin_inset Formula $\det\left[\begin{array}{cc} a & b\\ c & d \end{array}\right]=ad-bc$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula $\vec{a}\times\vec{b}=\left|\begin{array}{ccc} \vec{i} & \vec{j} & \vec{k}\\ a_{1} & a_{2} & a_{3}\\ b_{1} & b_{2} & b_{3} \end{array}\right|$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula $A_{i,j}\coloneqq A$ \end_inset brez \begin_inset Formula $i$ \end_inset te vrstice in \begin_inset Formula $j$ \end_inset tega stolpca \end_layout \begin_layout Standard \begin_inset Formula $\det[a]=a$ \end_inset , \begin_inset Formula $\det A=\sum_{k=1}^{n}\left(-1\right)^{k+1}a_{1,k}\det A_{1,j}$ \end_inset \end_layout \begin_layout Standard Razvoj po \begin_inset Formula $i$ \end_inset ti vrstici: \begin_inset Formula $\det A=\sum_{j=1}^{n}\left(-1\right)^{i+j}a_{ij}\det A_{ij}$ \end_inset \end_layout \begin_layout Standard Razvoj po \begin_inset Formula $j$ \end_inset tem stolpcu: \begin_inset Formula $\det A=\sum_{i=1}^{n}\left(-1\right)^{i+j}a_{ij}\det A_{ij}$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula $\det$ \end_inset trikotne matrike: \begin_inset Formula $\prod_{i=1}^{n}a_{ii}$ \end_inset \end_layout \begin_layout Standard Trikotna matrika ima pod ali nad diagonalo same ničle. \end_layout \begin_layout Standard \begin_inset Formula $\det\left(P_{ij}A\right)=-detA$ \end_inset , \begin_inset Formula $\det\left(E_{i}\alpha A\right)=\alpha\det A$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula $\det\left(E_{ij}\alpha A\right)=\det A$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula $\det\left(AB\right)=\det A\det B$ \end_inset \end_layout \begin_layout Standard Za možne napake ne odgovarjam. Srečno! \end_layout \begin_layout Standard \begin_inset ERT status open \begin_layout Plain Layout \backslash end{multicols} \end_layout \end_inset \end_layout \end_body \end_document