571_1.2

# 571_1.2 - 1.2 ROW ECHELON FORM KIAM HEONG KWA p 13 A matrix...

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1.2 ROW ECHELON FORM KIAM HEONG KWA p. 13 A matrix is said to be in row echelon form if (1) the ﬁrst nonzero entry in each nonzero row is 1 ; (2) if the k th row does not consist entirely of zeros, the number of leading zeros 1 in the ( k + 1) th row is greater than the number of leading zeros in the k th row; (3) all zero rows are below all nonzero rows. pp. 16-17 A matrix is said to be in reduced echelon form if, in addition to being in row echelon form, the ﬁrst nonzero entry in each nonzero row is the only nonzero entry in its column. The process of using elementary row operations to transform the augmented matrix of a linear system into one in row echelon form and into one in reduced row echelon form are called Gaussian elim- ination and Gauss-Jordan reduction respectively. Example 1 (Exercise 1.2.2(a) in the text) . Consider the augmented matrix x 1 x 2 = rhs 1 2 4 0 1 3 0 0 1 in row echelon form. The corresponding system is inconsistent be- cause there is no ordered pair ( x 1 ,x 2 ) such that 0 x 1 + 0 x 2 = 1 . Example 2 (Exercise 1.2.2(c) in the text) . Consider the augmented matrix x 1 x 2 x 3 = rhs 1 - 2 4 1 0 0 1 3 0 0 0 0 Date : June 18, 2011. 1 The leading zeros of a row vector are the collection of all zero entries preceding all nonzero entries. For instances, the row vectors (1 , 2 , 3 , 4 , 5) , (0 , 1 , 2 , 0 , 3) and (0 , 0 , 0 , 1 , 0) have no leading zeros, one leading zero, and three leading zeros respectively. 1

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2 KIAM HEONG KWA in row echelon form. The variables in the associated linear system corresponding to the ﬁrst nonzero elements in each row of the aug- mented matrix are called the lead variables ; the remaining vari- ables are called the free variables
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571_1.2 - 1.2 ROW ECHELON FORM KIAM HEONG KWA p 13 A matrix...

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