\chapter{Lineare Algebra} \section{Allgemeines} \subsection*{Definitionen} \subsubsection*{Levi-Civita-Symbol:} \begin{math} \varepsilon_{12\dots n} = 1 \\ \varepsilon_{ij\dots u\dots v\dots} = -\varepsilon_{ij\dots v\dots u\dots}\\ \varepsilon_{ij\dots u\dots u\dots} = 0 \\ \levicivita{i,j,k} = \begin{cases} +1, & \mbox{falls }(i,j,k,\dots) \mbox{ eine gerade Permutation von } (1,2,3,\dots) \mbox{ ist,} \\ -1, & \mbox{falls }(i,j,k,\dots) \mbox{ eine ungerade Permutation von } (1,2,3,\dots) \mbox{ ist,} \\ 0, & \mbox{wenn mindestens zwei Indizes gleich sind.} \end{cases} (\vec{a} \times \vec{b})_i = \sum_{j=1}^3 \sum_{k=1}^3 \levicivita{ijk} a_j b_k \\ \vec{a} \times \vec{b} = \levicivita{ijk} a_j b_k \vec{e_i} = \levicivita{ijk} a_i b_j \vec{e_k} \\ \det A = \levicivita{i_1 i_2 \dots i_n} A_{1i_1} A_{2i_2} \dots A_{ni_n} \end{math} \subsubsection*{Kronecker-Delta} $\krondelta{i,j}= \begin{cases} 1 & \mbox{falls } i=j \\ 0 & \mbox{falls } i \neq j \end{cases}$ \\s Die $n\times n$-Einheitsmatrix kann als $(\krondelta{ij})_{i,j\in\{1,\ldots,n\}}$ geschrieben werden. \section{Matrix-Operationen} \subsection*{Inversion} \begin{math} \hypertarget{fs_mtrx_inv_2d}{A^{-1} = \inlinematrix{a & b \\ c & d}^{-1} = \frac{1}{ad - bc} \inlinematrix{d & -b \\ -c & a}} \end{math}