Transcription of ELEMENTARY LINEAR ALGEBRA - Number theory
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ELEMENTARY LINEAR ALGEBRA - Number theory
ELEMENTARYLINEAR ALGEBRAK. R. MATTHEWSDEPARTMENT OF MATHEMATICSUNIVERSITY OF QUEENSLANDC orrected Version, 27th April2013 Comments to the author at 1 LINEAR Introduction to LINEAR equationsAlinear equationinnunknownsx1, x2, , xnis an equation of the forma1x1+a2x2+ +anxn=b,wherea1, a2, .. , an, bare given real example, withxandyinstead ofx1andx2, the LINEAR equation2x+ 3y= 6 describes the line passing through the points (3,0) and (0,2).Similarly, withx, yandzinstead ofx1, x2andx3, the LINEAR equa-tion 2x+ 3y+ 4z= 12 describes the plane passing through the points(6,0,0),(0,4,0),(0,0,3).Asystemofm LINEAR equations innunknownsx1, x2, , xnis a familyof LINEAR equationsa11x1+a12x2+ +a1nxn=b1a21x1+a22x2+ +a2nxn= +am2x2+ +amnxn= wish to determine if such a system has a solution, that is tofindout if there exist numbersx1, x2, , xnwhich satisfy each of the equationssimultaneously.
1.2. SOLVING LINEAR EQUATIONS 5 For example, with p = 7, we have 3 ⊕ 4 = 7(mod7) = 0 and 3 ⊗ 5 = 15(mod7) = 1. Here are the …
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