https://mathwiki.cs.ut.ee/ 2026-05-15T09:08:52+00:00 https://mathwiki.cs.ut.ee/ https://mathwiki.cs.ut.ee/lib/exe/fetch.php?media=wiki:dokuwiki.svg text/html 2014-12-13T22:03:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://mathwiki.cs.ut.ee/doku.php?id=linear_algebra:01_matrices&rev=1418508237&do=diff 1. Matrices A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns and surrounded by brackets. For example $\begin{pmatrix}2 & 6 & 3 & -5\\15 & 8 & 1 & 14\\2 & 2 & 2 & -8\end{pmatrix},$ $\begin{pmatrix}\cos\alpha & -\sin\alpha\\ \sin\alpha & \cos\alpha\end{pmatrix},$ $\begin{pmatrix}e & 3e\\0 & e\\ \frac{1}{2}e^5-\frac{1}{2}e & e^5 \end{pmatrix},$ $\begin{pmatrix}x^2\\y^2\\z^2\\w^2\end{pmatrix}.$ The size or dimension of a matrix is defined as $m \… text/html 2014-01-20T10:38:24+00:00 Anonymous (anonymous@undisclosed.example.com) https://mathwiki.cs.ut.ee/doku.php?id=linear_algebra:02_equations_and_matrices&rev=1390214304&do=diff 2. Systems of linear equations and matrices A system of linear equations is a collection of linear equations involving the same set of variables. For example, $$ \begin{cases} 5 x_1 + 2 x_2 = 4\\ 3 x_1 + 6 x_2 = 7 \end{cases} $$ is a system of two linear equations, both of then involving two variables $x_1$ and $x_2$ (the unknowns). The numbers $\ 5,\ 2$$\ 3,\ 6$$\ 4$$\ 7$$m$$n$$(\ast)$\begin{align*} \left\{ \begin{aligned} a_{11}x_1 +\, a_{12}x_2\, +\, \dots\, +\, a_{1n}x_n&=b_1\\ a_{21}x_1 … text/html 2014-12-13T22:12:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://mathwiki.cs.ut.ee/doku.php?id=linear_algebra:03_gaussian_elimination&rev=1418508778&do=diff 3. Gaussian elimination algorithm Gaussian elimination algorithm allows us to solve systems of linear equations. Let us have a system of linear equations: $$ \begin{pmatrix} a_{11}&a_{12}&\dots &a_{1n}\\ a_{21}&a_{22}&\dots &a_{2n}\\ \vdots & \vdots &\ddots&\vdots\\ a_{m1}&a_{m2}&\dots &a_{mn} \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\ \vdots\\ x_m \end{pmatrix}= \begin{pmatrix} b_1\\ b_2\\ \vdots\\ b_m \end{pmatrix} $$ and its augmented matrix: $$ \left(\begin{array}{rrrr|r} a_{11}&a_{12}&… text/html 2014-01-20T11:39:37+00:00 Anonymous (anonymous@undisclosed.example.com) https://mathwiki.cs.ut.ee/doku.php?id=linear_algebra:04_invertible_matrices&rev=1390217977&do=diff 4. Solving systems of linear equations via invertible matrices There is a special matrix denoted by $I$ and referred to as the identity matrix. It's number of rows equals to it's number of columns, and it has ones down the main diagonal (from upper left cornet to the lower right corner) and zeroes elsewhere. For example,$\begin{pmatrix}1\end{pmatrix},\ \begin{pmatrix}1&0\\0&1\end{pmatrix},\ \begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}$$1\times 1$$2\times 2$$3\times 3$$A$$m\times n$$I_n$$n\t…