# Systems of Linear Equations Notes - 16-1 16 Systems of...

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March 31, 201316-116. Systems of Linear Equations1Matrices and Systems of Linear EquationsAnm×nmatrixis an arrayA= (aij) of the forma11· · ·a1na21· · ·a2n...am1· · ·amnwhere eachaijis a real or complex number.We sometimes call such an arrayAanmbynmatrix.The matrix hasmrows andncolumns. The numbersmandnare calledtherow dimensionandcolumn dimensionofA.For 1im,1jn,, them×1 matrixa1j...amjis called thej-thcolumnofAand the 1×nmatrix(ai1. . . ain)is called thei-throwofA.IfA= (aij) is anm×nmatrix, then itstranspose, denotedATis then×mmatrix defined byAT= (atij) =ajifor eachi, j.If the row and column dimensions of the matrixAare equal, then wecallAasquarematrix, and we call the common value of its row and columndimensions, itsdimension.We will see that square matrices have specialproperties.We can addm×nmatrices as follows. IfA= (aij) andB= (bij), thenC=A+Bis the matrix (cij) defined bycij=aij+bij.We can only multiply matricesAandBifAism×nandBisn×p.That is, the number of columns ofAis the same as the number of rows ofB.
March 31, 201316-2In that case, ifA= (aij) andB=bjk, thenC=A·Bis them×pmatrixC= (cik) defined bycik=nXj=1aijbjk.Thus the elementcikis the dot product of theithrow ofAand thejthcolumn ofB.Both the operations of matrix addition and matrix multiplication areassociative. That is,(A+B) +C=A+ (B+C),(AB)C=A(BC).Multiplication of matrices isnotalways commutative, even for squarematrices. For instance, ifA="1101#andB="1011#,then,AB="2111#andBA="1112#.Let us consider some matricesA, Band illustrate these concepts.Example 1A="2312#, B="3-12123#C=A·B="9413538#B·Anot definedAT="2132#Example 2A=2-131213-22, B=1-3-2
March 31, 201316-3C=A·B=-1-75AT=213-12-2312,(A.B)T=h1-3-2iFact.(A·B)T=BT·AT.Leteibe then-vector with zeroes everywhere except in thei-th positionand a 1 there. This is called thestandardi-th unitn-vector.Then×nmatrix whosei-th row consists of a 1 in thei-th positionand zeroes elsewhere is called then×nidentitymatrix, and is denotedIn(or simplyIif the context makes the size clear).For anym×nmatrixAwe haveImA=AIn=A.2Multiplication of matrices by row and col-umn vectorsLetpandnbe positive integers.Letu1,u2, . . . ,unbenvectors inRp, and leta1, a2, . . . , anbenrealnumbers.The expressionu=a1u1+a2u2+. . .+anunis calledthe linear combinationof the vectors{u1,u2, . . . ,un}with coefficients{a1, a2, . . . , an}.Any expression of the above form is calleda linear combinationof vectorsinRp.It is useful to note the following properties of matrix multiplication.
March 31, 201316-4LetA= (aij) bem×nand letB= (bjk) ben×p. Then, of course,Cism×p.LetCribe thei-th row ofCandAribe thei-th row ofA, thenCri=Ari·BSimilarly, ifCcjis thej-th column ofCandBcjis thej-th column ofB, thenCcj=A·Bcj
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