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z?DTPli*Z55a&40GGR4bnYV*KLnuwk;sHDLG;`}a?{Gj^H4WInGMB9FS#4tZjrruwN zl+bfOvyr<-pb3WY_tBFJ|(|a zOAQ=(ro?8^a+|-EKt4sJ;oCD9D#{p`2o?-yDcSv8kgZ&N*j!-KR|IX`F1fjIVq6RZ z5+|JA$~C#=1HLCXs5QBcWuxH+0d9^AlEh@FmUZdQEbDGN0N3FsYbEu+0??2T5A%-tb@e zn`X5V-05vgnG+KPm{rhyL%VZ`3UJ|l{IP{R8T)2uMC7mk7|_L)jCrOPml`pwjfA7< p+V9h1{^v^LWNwG&=5FTd{>9bO3Xe~aPk^6a5RaKzUPA%z{{Uj#Bu@YU diff --git a/semester1/linear-algebra/linAlg-janishutz.tex b/semester1/linear-algebra/linAlg-janishutz.tex index d7bd994..36f97a0 100644 --- a/semester1/linear-algebra/linAlg-janishutz.tex +++ b/semester1/linear-algebra/linAlg-janishutz.tex @@ -44,7 +44,7 @@ ``\textit{Lineare Algebra ist Kulturgut}'' \end{Large} - \hspace{3cm} - Robert Weissmantel, 2024 + \hspace{3cm} - Robert Weismantel, 2024 \end{center} \vspace{3cm} diff --git a/semester1/linear-algebra/parts/solving-sle.tex b/semester1/linear-algebra/parts/solving-sle.tex index adb7d46..33ef48b 100644 --- a/semester1/linear-algebra/parts/solving-sle.tex +++ b/semester1/linear-algebra/parts/solving-sle.tex @@ -2,9 +2,12 @@ \vspace{-0.5pc} \section{Solving Linear Equations} \label{sec:sle} -Put the system of linear equation's factors (i.e. for a linear equation $ax + by + cz = u$, we would put $a, b, c$ into the matrix and $u$ into the vector) into a matrix, where each row is an equation and the result into a vector $b$. Then, we solve $Ax = b$ by Gauss elimination. +Put the system of linear equation's factors (i.e. for a linear equation $ax + by + cz = u$, we would put $a, b, c$ into the matrix and $u$ into the vector) into a matrix, +where each row is an equation and the result into a vector $b$. Then, we solve $Ax = b$ by Gauss elimination. -\textbf{Gauss elimination}: Transform matrix $A$ into upper triangle matrix by performing row transformations (adding, adding scalar multiples, multiplying by scalar) on it. All operations performed on $A$ have to also be performed on $b$. Typically, write down both as a matrix with a dividing line between $A$ and $b$. Then solve by back-substitution. Gauss elimination succeeds iff $A \in \R^{m \times m}$ and $A$'s columns are linearly independent. (Runtime: \tco{m^3}) +\textbf{Gauss elimination}: Transform matrix $A$ into upper triangle matrix by performing row transformations (adding, adding scalar multiples, multiplying by scalar) on it. +All operations performed on $A$ have to also be performed on $b$. Typically, write down both as a matrix with a dividing line between $A$ and $b$. +Then solve by back-substitution. Gauss elimination succeeds iff $A \in \R^{m \times m}$ and $A$'s columns are linearly independent. (Runtime: $\tco{m^3}$) \vspace{-0.5pc} \subsection{Inverse} @@ -20,43 +23,46 @@ Put the system of linear equation's factors (i.e. for a linear equation $ax + by \subsection{LU-Decomposition} \label{sec:lu-decomp} \setcounter{all}{13}\shorttheorem \textbf{LU-Decomposition}: $A = LU$. $U$ upper triangle, result of Gauss elimination, $L$ lower triangle, $(E_1 \times E_2 \times \ldots \times E_n)^{-1}$. -Transformation matrices $E$ ($E \cdot A = A_1$): transformation is a single entry in lower triangle, where the $i$ and $j$ are the two rows involved and the value of $e_{ij}$ is the operation performed on the two. $L^{-1}$ for size $3$: Diagonal, $L^{-1}_{1,2} = -L_{1,2}$, $L^{-1}_{2,3} = -L_{2,3}$, $L^{-1}_{1,3} = -L_{1,3}$. +Transformation matrices $E$ ($E \cdot A = A_1$): transformation is a single entry in lower triangle, where the $i$ and $j$ are the two rows involved and the value of +$e_{ij}$ is the operation performed on the two. $L^{-1}$ for size $3$: Diagonal, $L^{-1}_{1,2} = -L_{1,2}$, $L^{-1}_{2,3} = -L_{2,3}$, $L^{-1}_{1,3} = -L_{1,3}$. When multiplying up to three different ones, copies values to multiplied matrix. -If it is impossible to decompose $A$ into $LU$ without row exchanges, we get $PA = LU$, where $P$ is a permutation matrix (indicating which rows have been swapped). Time complexity is improved significantly with this \tco{m^2}. +If it is impossible to decompose $A$ into $LU$ without row exchanges, we get $PA = LU$, where $P$ is a permutation matrix (indicating which rows have been swapped). +Time complexity is improved significantly with this $\tco{m^2}$. -\shortdef \textbf{Permutations}: bijective function $\pi$ on matrix; Reorders the input structure (i.e. vector or matrix); +\shortdef \textbf{Permutations}: bijective function $\pi$ on matrix; Reorders the input structure (i.e. vector or matrix); \shortdef \textbf{Permutation matrix}: $p_{ij} = \begin{cases} - 1 & \text{if } j = \pi(i)\\ - 0 & \text{else} -\end{cases}$ -\shortlemma $P^{-1} = P^{\top}$. + 1 & \text{if } j = \pi(i) \\ + 0 & \text{else} + \end{cases}$ +\shortlemma $P^{-1} = P^{\top}$. \setcounter{all}{18}\shorttheorem \textbf{LUP-Decomposition}: $PA = LU$, where $U = P\times L A$. $P_j = I$ if no row swaps are performed at each step and $P = P_m \times P_{m - 1} \times \ldots \times P_1$. Rewriting this as $A = P^{\top}LU$, we can simply solve an SLE using LUP-Decomposition \vspace{-0.5pc} \subsection{Gauss-Jordan-Elimination} \label{sec:gauss-jordan} -\textbf{Gauss-Jordan-Elimination}: Generalization of Gauss elimination to $m \times n$ matrices, still works similarly to Gauss elimination. We aim to find REF or RREF (see \ref{sec:matrices}). To describe, we say REF$(j_1, j_2, \ldots, j_r)$ or equivalently with RREF, where $j_r$ is the $r$-th pivot. +\textbf{Gauss-Jordan-Elimination}: Generalization of Gauss elimination to $m \times n$ matrices, still works similarly to Gauss elimination. +We aim to find REF or RREF (see \ref{sec:matrices}). To describe, we say REF$(j_1, j_2, \ldots, j_r)$ or equivalently with RREF, where $j_r$ is the $r$-th pivot. The solution is then in a vector, whose components are either $0$ or the $r$-th component of $b$. Example: \[ \begin{bmatrix} - 0 & 1 & 0 & 0 & 2 & 0\\ - 0 & 0 & 1 & 0 & 3 & 0\\ - 0 & 0 & 0 & 1 & 2 & 0\\ - 0 & 0 & 0 & 0 & 0 & 1\\ + 0 & 1 & 0 & 0 & 2 & 0 \\ + 0 & 0 & 1 & 0 & 3 & 0 \\ + 0 & 0 & 0 & 1 & 2 & 0 \\ + 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \cdot \begin{bmatrix} - 0\\b_1\\b_2\\b_3\\0\\b_4 + 0 \\b_1\\b_2\\b_3\\0\\b_4 \end{bmatrix} = \begin{bmatrix} - b_1\\b_2\\b_3\\b_4\\\textcolor{Green}{0} + b_1 \\b_2\\b_3\\b_4\\\textcolor{Green}{0} \end{bmatrix} \] -If the green marked entry in $b$ were not to be $0$, then the SLE would not have a solution. +If the green marked entry in $b$ were not to be $0$, then the SLE would not have a solution. \textbf{CR-Decomposition}: see \ref{sec:matrices} for exaplanation. \setcounter{all}{24}\shorttheorem is described there