{ "cells": [ { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [], "source": [ "x,y,z,t = var('x,y,z,t')\n", "M = matrix([[1,2,3],[2,-3,1],[1,1,0]])\n", "v = matrix([x,y,z]).transpose()\n", "F = M*v\n", "p = matrix([1,1,1]).transpose()\n", "b = M*p" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This first example shows the interpretation of the system $M\\mathbf{x}=\\mathbf{b}$ as the intersection of planes where \n", "$$\\begin{bmatrix}1&2&3\\\\2&-3&1\\\\1&1&0\\end{bmatrix}\\begin{bmatrix}x\\\\y\\\\z\\end{bmatrix}=\\begin{bmatrix}6\\\\0\\\\2\\end{bmatrix}$$\n", "is viewed as a system of equations\n", "$$\\begin{alignat}{7}\n", "& &&x &&+ 2&&y &&+3&&z &&=6\\\\\n", "&2&&x &&- 3&&y &&+ &&z &&=0\\\\\n", "& &&x &&+ &&y && && &&= 2\n", "\\end{alignat}$$\n", "The solution here is $x=1, y=1, z=1$\n" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false, "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "text/html": [ "\n", "