{"id":5258,"date":"2014-04-24T06:05:23","date_gmt":"2014-04-23T21:05:23","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5258"},"modified":"2014-08-01T18:29:38","modified_gmt":"2014-08-01T09:29:38","slug":"systems-of-linear-equations","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5258","title":{"rendered":"Systems of linear equations"},"content":{"rendered":"
\n

A set of equations having the form <\/p>\n

\\left. \\begin{array}{ccc} a_{11}x_1 + a_{12}x_2 + \\cdots + a_{1n}x_n & = & r_1 \\\\ a_{21}x_2 + a_{22}x_2 + \\cdots + a_{2n}x_n & = & r_2 \\\\ \\cdots\\cdots\\cdots\\cdots\\cdots\\cdots\\cdots\\cdots\\cdots & \\cdots & \\cdots \\\\ a_{m1}x_1 + a_{m2}x_2 + \\cdots + a_{mn}x_n & = & r_n \\end{array} \\right\\}\\cdots(16) <\/p>\n

is called a system of m linear equations in the n unknowns<\/em> x_1,\\ x_2,\\ \\cdots,\\ x_n. If r_1,\\ r_2,\\ \\cdots,\\ r_n are all zero the system is called homogeneous<\/em>. If they are not all zero it is called non-homogeneous<\/em>. Any set of numbers x_1,\\ x_2,\\ \\cdots,\\ x_n which satisfies (16) is called a solution<\/em> of the system. <\/p>\n

In the matrix form (16) can be written <\/p>\n

\\displaystyle \\left( \\begin{array}{cccc} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\cdots & \\cdots & \\cdots & \\cdots \\\\ a_{m1} & a_{m2} & \\cdots & a_{mn} \\end{array} \\right) \\left( \\begin{array}{c} x_1 \\\\ x_2 \\\\ \\cdots \\\\ x_n \\end{array} \\right) = \\left( \\begin{array}{c} r_1 \\\\ r_2 \\\\ \\cdots \\\\ r_n \\end{array} \\right) \\cdots (17)<\/p>\n

or more briefly AX = R \\cdots (18)<\/p>\n

where A, X, R represent the corresponding matrices in (17). <\/p>\n<\/blockquote>\n