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.

2 CHAPTER 1. LINEAR EQUATIONS Note that the above system can be written concisely as Xn j=1 aijxj = bi, i = 1, 2,···,m. The matrix a11 a12 ··· a1n a21 a22 ··· a2n

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  Linear, Number, Theory, Equations, Linear equations, Algebra, Linear algebra, Number theory

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